Since you posted a Melee video and I assumed this was the Melee forum when I first wrote it, everything that follows is written with Melee in mind.
General points that apply regardless of your video; do all of these things if you're not doing them already:
• Read Smashboards threads (check out the "tips for a new Luigi player" type threads first).
• Watch videos and pay attention; look for reasons why a player got punished, what strategies they do in certain situations and why those choices were good/bad, etc.
• Pay attention to your own games as you're playing. Like watching videos, if you get punished, think why and try to fix it. There are more detailed explanations of this kind of thing elsewhere on Smashboards.
• Ask the people you play to point out bad habits you have or things that you're doing well at.
• Theorycraft strategies and then test them out.
• Keep a good mindset (again, many resources for this).
You might be interested in Wobbles' blog, which covers these things and many more; it's very helpful and interesting:
http://www.compete-complete.com/
I should mention that I am still a scrub, but I will try to give the best advice I can. LuckiIy, my main (and by "main" I mean "only") training partner is a Roy main, and this matchup is very strongly in Luigi's favour. Luigi punishes Roy really hard and in this matchup Roy is really short of combos, kill setups, recovery ability, zoning ability, approaching ability, edgeguarding ability, and pretty much everything he needs in order to be threatening.
First of all, work on your tech skill. The sooner you get good tech skill the sooner execution will stop holding you back from implementing better strategies. Practice movement, making sure that it's smooth and consistent (don't jump when you're trying to wavedash, and make sure your wavedash can be long enough to move you about half the length of Final Destination). Use walks and dashes out of wavedashes and execute the next wavedash as quickly as possible to make your movement smoother (watch videos of Blea Gelo and Vudujin; they have the best movement of the top Luigis).
Walk forwards out of wavedash backwards is really good as it keeps your movement ambiguous and noncommittal (mix up the timings, lengths, and followups); some options out of it are things like wavedash forwards->forward tilt, wavedash forwards->grab->immediate throw (if you do this unexpectedly the opponent will often miss the DI), or dash-dance. This is comparable to dash-dancing in that it's a neutral game option that keeps your movement ambiguous and allows you to bait your opponent to approach (and then punish them for doing it).
Learn to ledgedash to greatly improve your ledge options (you can ledgedash into a buffer roll and be intangible the whole time, by the way). You shouldn't jump straight from the ledge (it's called the "tournament winner jump" for a reason); drop down and then jump instead to leave you able to act sooner.
It looks like you're trying to shield drop with all the spot dodges on platforms, so work on that (moving the control stick to the bottom corner and then rolling it to the bottom middle makes it easier; as well as the motion possibly being more comfortable, the area which produces a shield drop is actually larger at the side than in the middle). Don't roll too much, especially not when you're playing a character with ground movement as good as Luigi. Make sure to wavedash out of shield (if you're doing it after being hit then delay your jump, otherwise you'll input the jump during shieldstun and will end up either releasing your shield, spotdodging, or rolling instead); Luigi has the advantage of having his primary movement option (wavedashing) available to him out of shield.
You need to work on your recovery. As an example, at 33 seconds, you get hit with a forward smash (I assume the missile on stage was an accident; never do that when your opponent's right next to you), then you use your jump as soon as you can (you do this every time), and then you up-B when a tornado could have brought you close enough to have a chance of making it back.
When recovering, don't use your jump immediately; missile first and then, if you have to, use your jump when close to the stage. Every time you were recovering you used your jump straight out of hitstun. Without a jump Luigi is even more vulnerable than necessary, and missile covers enough distance that you often don't need to use your jump to be in up-B/airdodge/tornado range. Even then, it's safer to use the jump near the stage since his options without a jump have really linear trajectories (Luigi's recovery can be edgeguarded on reaction a lot of the time; you have to make it as hard as possible). Use your tornado to recover; this thread contains useful information on it:
http://smashboards.com/threads/rising-cyclone-ptp-frame-data-guide.384946/
You use a lot of fireballs in neutral, which can help to tack on damage and force the oppontent approach or give up stage control. Since the reward for this is usually very low and the risk very high you shouldn't use them as much as you do, but you can do it sometimes (generally, the slower, floatier, and campier the opponent, the more effective fireballs are). You use multiple standing fireballs which leaves you open; Roy can shield, jump over, or clank with them very safely and easily. You need to use a fireball while moving so that you can punish his attempt at avoiding it. Short hop and pull back if you need to. You do wavedash in with fireballs, but this is a terrible idea since they don't have very much hitstun, leaving you standing right next to an opponent at something like a 25 frame disadvantage (open to almost any punish the opponent wants, in other words), while the biggest reward you could possibly get out of it is 6% damage (even if the opponent does something pointless or bad like roll away). Onstage, you pretty much don't want to be using fireballs at all against Roy.
Fireballs can be used when edgeguarding; you can do things like putting a fireball in the opponent's path to force them to recover in a certain way (or more accurately, to reduce the viable recovery options available to them), or wasting their jump by hitting the out of it immediately after they use it. Fireballs can also be used while recovering; you can use them relatively close to the stage to interrupt Roy's edgeguard and get to the ledge or airdodge onstage before he can re-establish himself.
When you're going to approach, you wavedash back and forth a lot and in a very similar way each time (you wavedash back and forth then wavedash forward and attack immediately, all with wavedashes of the same length), this and the lack of fluidity in your movement makes your approaches predicable; mix up distances and timing. You approach with wavedash fireballs (previously covered) and wavedash down smash a lot. Wavedash down smash is good, but it gets predictable like anything else that's used frequently. You also have lots of other options such as wavedash grab/jab/>grab, wavedash forward tilt, SHFFLed nair, tornado, wavedash up tilt, wavedash up smash, etc.
It looks like you're wavedashing up to your opponent and then doing your attack while your momentum is almost gone, or else just doing standing down smashes; Luigi is much safer while he's moving. If you do an attack like that and the opponent shields it or otherwise avoids it then you're stuck in place and unable to act, giving your opponent an easy punish most of the time. If you're moving, you can put distance between you and your opponent so that they have to move further to punish you, giving you the chance to get out of lag and escape or continue pressure (or else forcing them to use a less rewarding option).
When you're wavedashing in with an attack, try to sometimes do cross-ups (wavedash far enough so that you move through them as you attack); it makes you harder to punish. If you wavedash at high speed through someone's shield and hit them with an up smash or the second hit of down smash from behind then not only will they be unable to shield grab you (because they're facing the wrong way), but the shieldstun and pushback will be so great that they won't be able to hit you with any of their other out of shield options either.
Roy has to play this matchup really defensively; he should be dash dancing a lot and going for grabs, down tilt/side-B pokes, and sometimes spaced aerials (you can just crouch-cancel these for a really long time so they're not such a great option). Up/down throw and down smash are your main combo starters. Up throw leads to good combos, right up until kill percentages if the opponent doesn't DI it very well. Often you'll be wanting to up or down throw into a single aerial or just forward throw to establish positional advantage.
You can go for forward throws off the stage to set up edgeguards from low percentages (you can use wavedash forward tilts in a similar way which creates a mixup, although they can be crouch-cancelled/ASDIed down for a long time). Comboing into kill moves is easy and can be done from pretty much anything; I particularly like soft nair (it gets weaker the longer it stays out) to forward/down air, since it has enough hitstun to combo straight into an aerial but doesn't send them far even at high percentages (and is difficult to DI out of).
Tornado is pretty safe, if low reward, but there are usually better options; it clanks with jab, forward tilt, the first hit of side-B (and others but I'm not sure which), down smash (both hits), forward smash (only the tip), and up tilt; it does not clank with down tilt (since the Tornado hitboxes are too high up to cover Luigi's hurtbox, I think). Depending on percent and DI, it is possible but difficult and rare to connect with both hits and then tech chase off it; you may have to mash the B button a little to reach with the second hit (using the quick, low jump that he does). If you hit with the first hit early then you can continue to move through the opponent to put yourself out of range of a counterattack. It should be used situationally; you can use it more often if the opponent is unfamiliar with the matchup. Most of the time you're using it to disrupt them or as a tech-chase mixup and you shouldn't expect to get much off of it. That said, if they miss their tech (often due to being caught off guard or landing on a platform in an awkward way), thenyou can convert off it pretty well.
Roy is very vulnerable to crouch cancelling on pretty much all of his moves until high percentages; down smash or grab out of crouch cancel to punish with a combo. You should be crouch cancelling pretty much everything except forward/down smash until extremely high percentages.
Edgeguarding Roy is very easy; you don't need to try to kill him, just cover his (extremely limited) recovery options. If you're going to grab the ledge (this is what you should do when he's forced to sweetspot) you can stall/wait before grabbing it to maximize intangibility frames. You can drop/jump from the ledge with a bair (usually; other aerials can be used too) to knock him away and try to take his jump while forcing him into a more disadvantageous position. If you do this immediately after grabbing the ledge you can hit them with the bair while you're still intangible (this is a good option if they're recovering low).
If you're on the ledge and he lands on stage with his up-B (since his horizontal distance is so poor he can't get very deep into it) then you have plenty of time to ledgehop, hit him with an aerial, and then repeat the edgeguard (even if you knock him to the other side of the stage instead of the way he came, you can usually get there quickly enough to edgeguard him again). At very high percentages you can also use your regular stand/waveland onstage and then up-B to kill (like when edgeguarding Sheik).
If he's within his up-B range or recovering high and you can't grab the ledge quickly enough or don't want to deal with the onstage recovery mixup, then wavedash forward tilt (this covers overshot sweetspots), or drop offstage/jump up and hit him with an aerial to either knock away or stage-spike him (be ready for the tech if you stage spike him). He can't do much against this when recovering low as he will die if he uses any aerials offstage at that height, and he isn't close enough to airdodge onstage; the best he can do is try to up-B through you. Be aware of the airdodge onto platform option and be ready to hop up with an aerial or something of similar effect.
Roy cannot edgeguard Luigi very well, nor combo into his kill moves well. He also has very few kill moves; neutral aerial is reliable but doesn't kill until high percentages, forward smash is powerful and kills at relatively low percentages but difficult to safely connect with against Luigi if he remains mobile and spaces well (and is difficult to combo into at kill percentages), and down smash can kill off the top but is very punishable on shield and on whiff.
DI his forward throw properly (down and away) to stop him getting easy damage/forward smashes. Bad DI of the forward throws at low percentages gives him guaranteed regrabs. When Roy edgeguards you with forward smash you should be able to DI it such that you can recover high, giving you a lot more options. As long as you keep your jump and height, you should have a pretty easy time of getting back onto the stage.
Despite Roy's terrible offstage edgeguarding, without your jump he can still kill you fairly easily because of how much height you lose moving back towards the stage after being knocked away, so don't waste it. When recovering high, you are threatened by his forward smash and Flare Blade (neutral-B), which will generally steal your jump and knock you away, leaving you below the stage or forced to missile close to it; this leaves you with fewer options for Roy to cover. Regadless, you have more options when you are able to recover high. If you're recovering high then you can edge cancel missiles on the platforms or airdodge onto them (or the stage). Eddy Mexico is without doubt the Luigi that makes the most use of edge-cancelled missile recoveries, so watch videos of him.
Unlike most of Roy's moves, his Flare Blade has the sweetspot on the tip of the sword (which is one of the reasons it's a good edgeguarding tool). If you miss the ledge sweetspot then he can forward tilt you. He can also occasionally nair or fair you away. When he back throws you at the ledge he gets a free forward smash at if you DI it inwards or jump, but he doesn't get anything if you DI away.
If you ledge tech Roy's forward smash during your up-B, then you can always up-B again and grab the ledge before he can hit you with another forward smash (you can fast fall after the tech to make yourself safer). Every time you try to sweetspot the ledge with up-B, you should always ASDI or SDI into the stage and hit the tech button; there is no downside to doing this, since if you are not hit with anything then nothing will happen, and if you do get hit then you will tech provided that the ASDI was sufficient. Just be certain not to be holding up on the control stick as you tech, since you will end up doing the wall jump tech which will force you to missile back to stage in a very easy to intercept manner. Furthermore, because the tech window is so large, the variance in tech timing between different moves is so small (if there is any at all), and the height of a sweetspot p-B is always the same, you only need to learn one timing; you can use training mode to practice this by spawning proximity mines and then Z-drop them as you pass the ledge.
When you're getting up from the ledge, you have plenty of options and mixups are important: ledgedash->various options (buffer roll, down smash, etc.), stand (press forwards), ledge roll, ledgehop aerial (you can hit with any aerial while retaining intangibility, but Roy can outspace this with a forward smash after the intangibility has expired and steal your jump while setting up an edgeguard), fast fall down->regrab ledge, getup attack, tornado, and fireball, to name most of them.
This information is barely useful but I want to put it somewhere: when getting hit by the start of Roy's up smash while grounded, you can crouch cancel tech and punish, but this stops working at low percentages (about 30%). However, when you get hit by the first hit you can automatic smash DI down while regular DIing horizontally away (by holding the C-stick down and the control stick right/left) and then tech to escape immediately (taking only 1% damage) even at well over kill percentages.
That's all I can think of right now. I'll edit in more if I think of it. My Roy matchup explanations are missing important things. I feel that Roy is the kind of matchup without many gimmicks to it (unlike some other low tier matchups like Game & Watch or Bowser); just play a normal, solid game and you'll out-character him. Be patient and don't overcommit to anything; your reward for getting into any kind of advantageous situation is higher than his, so you'll be able to win by attrition if you play safely.
I did a lot of the things you were doing when I first started playing competitively; you will definitely improve with experience, so don't feel bad about losing (you're always going to be losing to someone unless you're the best player by a wide margin; it's motivation for improvement). The most important thing is having a good mindset so you can keep trying and being aware of your problems so you can work out how to fix them.
