Man I'm gonna have to make another list for this:
U-tilt: When your opponent is within the 130%+ range and you juuuuust can't seem to kill them, U-tilt is perfect. Just enough knockback to kill them from off the edge or otherwise, and fairly quick with good range.
Knee: The knee isn't something you can truly rely on, but if you hit with it at 90%, your opponent's in trouble.
F/D-smash: I prefer D-smash instead of F-smash. Try spamming it in training. It has good knockback, great shield pushback, ok range, hits on both sides, and is less laggy than F-smash before and after. Also correct me if I'm wrong but I think it has more knockback than F-smash anyway. Anyway if you use it too much it you won't be able to use it as a ground kill move though. It's up to you.
Back-air: Back-air sweetspotted is much more reliable than the knee. It comes out fast and has pretty good knockback, but they have to be at a pretty high percentage for it to work.
Falcon Punch: USE MDNIGAEMS
Captain isn't really a pure killing character, though. His moves all have just enough knockback to send the opponents off the edge, though. As it's been stated already in this thread, I'm pretty sure your best bet with Falcon is to edgeguard/spike/gimp off the edge. Up-airs while the opponent's coming back to the stage are awesome when you still have your double jump. All the above mentioned moves will easily get the opponents off the stage, including f-throw and b-throw, which can lead to another off-the-edge-aerial.