As a DK player and a bit of a Link I think I can offer a good bit to this discussion. Here's the points we need to discuss...
Common things you'll see from them
Bair, bair, bair. Not all DK's use it as bread and butter, but enough do. They'll do short hop bair, short hop double bair, empty double jump-fastfall bair, and so on. They camp it. And Link, sadly, has not a lot of options for outspacing this beast. Good DK players will buffer their landings into reverse utilts, which are rock-solid for shield pressure and can still outspace most of Link's arsenal. Every time you land, prepare for him to charge an fsmash or dsmash, as the massive Verizon coverage and long hitbox time for both of these options are perfect for catching you in a fall-trap. Despite DK having a large variety of fun and unique styles (which is why I love him), you won't see too much difference in a tournament-setting in terms of common tactics. Apart from bair and utilt spacing, and landing- or airdodge-traps, it's just a grab-bag of playstyle with his collection of situational moves. However, though it isn't 'common', DK's cargo throw is his deadliest weapon in this match, and unless you have brilliant spacing, you'd better hope he doesn't use it a whole lot. Link is one of the very few characters in the game (off the top of my head, Olimar and Wario without his bike) that he can kill at basically ANY percent with a simple cargo dthrow as soon as he runs off of the edge. This is huge for him because usually his gimps come from his cargo stage-spikes, which are not only limited to certain stages, but can be teched and he can be counter-gayed. But he can just grab Link on his hairy back and toss him off, grab the edge, up-b stall a few times, and it's over.
How do we deal with their moves
I can't give Link a whole lot of suggestions here. The main advantage you have is projectiles - use them. Though a good DK will bair through a lot of your ranged options, I believe bombs will trade (AKA explode on his foot). Projectiles wouldn't be too much of an advantage if it wasn't for DK's shield, which is exposed to a lot of pressure and pokes/stabs, which Link's projectiles are great for. Another good option is your extra-meaty dair, which is risky (as we all know) but synergizes with DK's weak shield very well for you. Unless he's at full strength, it's often going to find a hole. Even if he stays off the ground to cover this option, surprise sword-plants (as scrubby as they are) will simply beat everything DK has in the air, and often times when he tries a uair juggle or trap, you can just pound through it with your falling dair.
How they deal with our moves
If you space with anything other than projectiles, he'll outspace it with his bair, consistently. If you spam your projectiles, he'll try to double-jump over them and space a bair during your lag (which you should be able to punish, with a couple mindgames). If you go for grabs, he'll try to spotdodge them and punish with a semi-charged smash (or a fwd-b to smash)... but his spotdodge animation is so long, you're relatively safe. I don't know Link's core playstyle well enough to offer a lot of suggestions here, however.
Stages, good/bad neutrals, good/bad counterpicks
Donkey Kong has some very, very strong stages, and they apply to this matchup. My personal favorite, and something to fear, is Pokemon Stadium 1, which not only gives him the best and fastest cargo spike in the game (which WILL insta-gib you, with less wiggle time), but a few more great benefits, like ground-pound combos on the windmill, platforms at a nice height, and a very low ceiling, perfect for his KO moves. His other strongest counterpicks are undoubtedly Delfino Plaza and Brinstar. On Brinstar, he has a lot of potential to autocancel his up-b, and his smashes gain extreme hitbox length when applied on any of the destructible elements (he also has nice lava combos, and a trick where he can cargo you into the lava, and you'll fall down to your death, and he'll pop up with the lava). On Delfino, he has some rage-worthy stuff, including auto-waterspikes from a simple cargo release, wall infinites from the same technique, some of the fastest walk-off kills in the game, and a surprisingly scary sharking game on the regular stage (he can drop off, poke any aerial through the floor, and up-b to either side safely).
When it comes to neutrals... there are some important factors. He can utilize all of them, due to some interweaving of benefits. First is the ability to cargo spike, which is less important as he can gimp you without it. Regardless, those neutrals are FD, Lylat, BF, and SV. Only YI takes that option away, and some of the best DK players actually favor YI over any other stage or counterpick on the entire game due to a strict-timing trick where he can up-b onto the stage, moving along the ground, and gain 100% invincibility during the full length of the move. I personally hate YI because of every other aspect of it for DK, but many swear by it and be prepared for it. Platforms are also a large advantage for DK, which his many options with his large range to hit you while you occupy one. For this reason, and your projectile game, I would recommend FD as your neutral to strike or random to. As for your counterpicks, I'd be lost, because his bad ones are often your worst (like Rainbow Cruise).
Strengths and Weaknesses
For this matchup...
DK
+ Better range
+ Gimp potential
+ Large KO power
+ Safe recovery
+ Heavy with DI options
- Weak shield
- No projectiles
Link
+ Projectiles
+ Shield-stabbing attacks
+ KO power
- Easy to gimp
- Mid-to-low range
