I agree with most of Elven's points here. I would DEFINITELY start FJ pilling all over a Falco; if you can time it right you can go right over his lasers that way and start seriously crippling him with some % and interrupts of his movement. Caping is a good idea with the lasers when done tastefully, though I wouldn't reccomend going out of your way to do it or doing it straight from the ground; this get's kind of risky and leaves you stationary. As for the dashdancing point, meh, you -could- dashdance, but I've always found a ****load of wavedashing to work a bit better with doc, anyway. Dashdancing is better when you're reading techs for grabs, etc. As for reading techs and grabs, uhhh... definitely try to do some more grabbing, Chaingrabs or otherwise. Doc's got some nasty stuff out of grabs against falco.
I, for one, would encourage you to use the cape and not argue that it'll get you ****ed up, but you do have to use it wisely. Try to only use it in the air (it can be wavelanded this way, which I think takes alot of the risk out) and only when you -know- other modes of evasion or attack are inconvenient. Also, I think it's reasonable to use in recovery, but not to help you gain further ground. It stops your DI short, so often times it can mess you up. In disagreement with Elven, though, it's not going to straight up screw you over, so I wouldn't listen to that. It isn't, however, an awesome recovery move. What it can do for you is add some timing mindgames, delaying your recovery a frame or two so you can pop up -after- your opponent has jumped the gun and thrown out their ledeguarding attack. This, again, must be used wisely, and if done too often, will just have your opponent waiting you out every time. It's a mindgames tool here, not a legit recovery technique.
Your doc looks reasonably fast, which is awesome. I'm tempted to ask if you've always been a Doc main, since your tech seems pretty on, but alot of doc-specific things are pretty off. I'd recommend checking out the Doc Q&A forums to look at alot of Doc combos against certain matchups and how to play him therein. You seem to know how to move a character around, but not how to use his quirks to really **** someone up. Also, yeah, try to use the DJ more sparingly, since it might get you pwned, especially against someone who's as aggressive as Falco.
I think it looks like a pretty **** good aspiring doc though; main things to look at are what Doc CAN do that other character's can't. Good luck, keep up the good work. : )