I composed my own tier list, and it looked like this:
S Tier:
1. Fox, Falco
I put them tying for best character because their archetype is pretty much the same (given that they are clones), but they just operate slightly differently; Falco has flinching lasers, a shine that opens up to a world of combo trees, a faster side B which makes his recovery harder to react to, and the best Dair in the game by far. Meanwhile, Fox is extremely fast, has very fast damaging lasers, a shine that gimps nearly every character in the game, a very long recovery, and very consistent vertical KOing moves.
3. Jigglypuff
I took the time to rethink Jigglypuff's placing and read articles on why M2K thought that it was #3 in the game, and judging by Hbox's performance at EVO, it's very clear that Jigglypuff is pretty much meant to take the lead, and keep it by being campy (this is apparently the only reason M2K has succeeded whenever he uses Jigglypuff, so it's not surprising Hbox would adopt the planking strat). Jigglypuff can seriously cripple any momentum its opponent has with a rest or a good Wall of Pain, and it is very hard for most characters to take the lead back; in fact, this probably affects more characters than Sheik's chain grab. Jigglypuff takes the 3rd spot, easily.
4. Sheik
Sheik is the next obvious choice here, with her being a low tier devourer and having moves that naturally combo into each other (pretty much all tilts) and gimp the opponent (Fair, Dsmash). I would say one of the only reasons she is above Marth is due to needles, which have a larger range than Marth's sword and negates far more powerful projectiles due to having actual hurtboxes.
5. Marth
Marth simply cannot be overlooked no matter what version of the game you play. He will always have the top 5 spot, but sadly, in NTSC, I think he should take the lowest spot of that top 5, simply because range is all he has against the rest of those characters. He doesn't have the air mobility of Jigglypuff nor dash speed of Fox, but Marth can still definitely do work against every character with good zoning and proper punishes.
A Tier
6. Peach
Peach is not a weak character at all (unless you're a Jigglypuff player...). Her edgeguarding game, float cancels, and turnips can really frustrate even the gods of Melee if they're not careful. She is a little on the slow side, but her combo game rivals that of Falco's at times if you find the right opportunity. No new player should be afraid of considering Peach. She is a monster in the right hands.
7. Ice Climbers
With an infinite on every single character, you'd think that Popo and Nana would be at the top of the list... however, they do have to be together to pull it off, and losing one of the two makes the ICs very weak mid stock. Still, the handoffs and Wobble are something to look out for.
8. Captain Falcon
I don't really like the way Falcon plays, even incorporating 20GX. It's very read heavy and the optimal gameplan is often something that the characters above him just simply do better. He's still ranked pretty high considering his combo game into Knee is no joke, but the way you land them requires very hard reads to do so at top level, or so it seems at least for me.
9. Pikachu
Pikachu is really just like a poor man's Fox; he has decent speed, great vertical KOing moves, good gimping moves, and a really long recovery (honestly probably the best after Jigglypuff). The only downside is his awful, AWFUL range on his attacks. But he has the tools to take on pretty much any character.
10. Samus
Samus is, apparently, the bane of spacies, being pretty much the only other "mid-tier" to stand an even chance against them. Her missiles work similarly to Falco's lasers but with decent KO potential, she has really good range (dat Zair), and has that really useful up B OoS. If only she hit a little bit harder and didn't have such a high short hop.
11. Yoshi
Yoshi should really be higher on the list, no matter what anyone says. He probably shouldn't be this high, but he is literally the only "low tier character" who has made a dent in any major in the past year. Heavy armor on double jump, shield shenanigans like parrying, and amazing platform movement potential are not things you want to mess with.
B Tier
12. Dr. Mario
Dr. Mario is a surprisingly great edgeguarder with Bair and cape, and he has decent KO moves to boot. They are sadly, either a tad weak (Dsmash) or have awful range (Fsmash and Fair). His recovery is also notoriously atrocious. That might be the single reason Shroomed switched off of him since 2014.
13. Luigi
Luigi gets the middle spot, yay~ With the longest wavedash in the game and very fast moves, he can cause opponents quite some trouble if he remains unpredictable with his movement. However, his recovery, KO moves, and range are all slightly lacking. He really doesn't excel in any aspect aside from wavedash length, and this low traction also works against him because it makes shielded punishes much harder to do.
14. Mario
The Mario Bros. are packed really close together in this list... which is to be expected, since Mario and Luigi are brothers and Dr. Mario is, well, Mario in doctor garb and smarter about how to hit enemies. Mario has better range and recovery than Dr. Mario, but that's about it. His Fair is awful no matter which hitbox you land (even the meteor smash requires some serious sleeping on the opponent's part for them to get hit by it), his Dsmash is even weaker, and his sourspotted Fsmash when not sweetspotted is worse than that of Marth's (and you thought Marthritis was terrible...).
15. Ganondorf
Ganondorf hits hard and has decent range. That's about it, really. His attacks and movement are just too slow for him to be viable and useful against the top 5 characters. While you can make miracles happen, they do not come often or consistently enough to be a real game changer.
16. Young Link
The young Hylian Hero is a fast attacker, fast walker, and has a surprisingly good projectile game. Despite that, good opportunities to land his combos are scarce because they don't... LINK quite as often as you like. His recovery's also not that great, especially for someone his weight.
C Tier
17. Link
The older Hero of Time is bigger, bulkier, and a little stronger... but his weaknesses are just like Young Link's, but exacerbated due to his lower speed. They can both moonwalk, though. That's something, right?
18. Donkey Kong
If there was anything close to a Wobble in the lower tiers, it would be Donkey Kong and his cargo throw, which wreaks havoc on spacies should they ever end up in the ape's arms. But aside from a decent Bair and Uair, that's pretty much it. The tie being a hurtbox on his body and the extremely small shield only causes more problems for being so big and relatively slow.
19. Roy
I think that because Roy is a clone of Marth, he's actually an ok character. It's just that his moves really, REALLY suck when they're sourspotted. There is no reason for Roy's moves to have such a disparity in strength between the sweetspot and sourspot. He may have that grab range and up throw, but seriously, poor Roy.
20. Mr. Game & Watch
A 9 is literally the only thing he can do to save himself from the bottom tiers. Oh, and he has a 1 frame up B that travels super far upwards. But his shield is just... WHY? He can't L-cancel all of his aerials, either. Makes you wonder if they finished this game on time.
21. Mewtwo
Mewtwo has a slower charge shot, strongest up throw in the game, and a very good recovery. But he's too light for his own good and has awful power in every other aspect. Range isn't great either, and like Donkey Kong's tie, having the entire tail as a hurtbox isn't fun.
22. Pichu
I'm taking a lot of pages from M2K's thoughts, but I really don't think Pichu is all that bad, either. His Nair does everything for him, and his smashes are really freaking strong. Like Link and his bulkiness and slowness, though, Pichu's range weakness is exacerbated from Pikachu's. Self-damage is also really bad, but most of it can be avoided by just using Pichu's side B and up B.
D Tier (aptly named, GMS)
23. Ness
Ness's recovery is really slow to execute and his aerials are much slower than in Smash 64 (except for Nair, Nair is one of his only good moves and it has very little range). His killing power isn't all that great, and his projectiles are even slower and not even that useful if they somehow hit. If Yo-Yo glitch could be a regular thing, he'd be the best character in the game, but alas, we have a shell of a young boy who didn't even do that well in the previous Smash game he was in. -pat pat-
24. Bowser
Up B OoS, 0-99% getup attack and SHFFL Fair are probably the only decent things Bowser has in his arsenal. You won't see Bowser land the most powerful Fsmash in the game often enough to make a significant impact.
25. Zelda
There must be a reason barely anyone wants to switch to Zelda, even just for the recovery. She is literally the slowest character in the game (dash, probably air speed, and non-tether grab). No combo game, the only things saving her are really fast neutral B and Dsmash, and really strong sweetspotted Fair and Bair.
26. Kirby
I'm really sorry, Kirby. You may be Sakurai's love child, but you got wrecked hard coming into Melee. The only thing you retained were your up smash, Bair, and really cute taunt. You have nothing else going for you. <\3