Falcon vs Link may seem like a hard matchup initially, but once you know it's various nuances, I think it's probably 60-40 in our favor.
Fighting Link effectively is all about patience, as reckless play will often result in getting juggled/techchased for days. You don't really want to approach Link head-on, try jumping over his projectiles, then fastfalling when he's farther away from you and slowly get closer to him. Once your close, try zoning with aerials (especially with Bair/Nair) to prevent him from grabbing you while pressuring him. Once you get a hit, you want to follow up with Up-Throw Short hop U-Air, if you can safely U-air again go ahead, if your not so sure, try dash dancing under him and maybe zone with more aerials until he's hit again or forced to land, which with your speed, lets you easily convert that into more hits or a grab. Most of your moves clank with his which is what makes aggressive zoning so effective against Link since your attacks are generally faster than his, you can convert random hits into combos much easier than he can due to his slow grab, and it's hard for Link for to start combos on you if you mixup your blazing fast movement with Nair/Bair walls. When being chaingrabbed, try quickly rolling the control stick around before you tech, so that not even you know what will happen when you tech, as this makes your teching patterns impossible to pick up on, since they're random. You also need to be patient against a more aggressive Link as his shffl'ed nairs can easily lead to large, painful combos. If he attacks you head on, you can easily shield grab him, if his attacks are more well spaced, you can easily falcon kick him, since it plows through or trades with almost everything he does, and he generally isn't fast enough to punish you for it, even if he shields it.
