I literally copy-pasted my own post from another thread, cuz I really don't want to type all of this up again, but here's some tips:
Bread and butter combos. Try approaching with SHFFL'd N-Airs -> Gentleman, but you have to know when to go for it and when to hold your ground or back off. Dash dancing is of utmost important, but that's the case for just about any character. Try mixing up dash dancing with moonwalking and wavedashing. It's very important to have high mobility and to have control over falcon as he is very fast, so put one of his best assets to use. U-Air is your other best friend, but use it more sparingly than your N-Air. U-Air is an excellent follow-up when initiating combos (i.e : Grab -> U-Throw -> U-Air [This simple combo varies on the character matchup as well as the enemy's percentage]). Grab -> D-Throw is also a great way to start up a combo on the characters that you shouldn't U-Throw. Try following those with N-Airs or U-Airs and finish with a knee when possible. A good way to understand Captain Falcon's combo game is to write all the names of his moves on pieces of paper and mix them up, throw them in a hat, and pull out a couple of them and put them in order, and there's your combo.
The knee is your most powerful killing move by far, but the B-Air is also pretty strong. U-Air can also be used to kill at very high percentages. Make sure you D-Air punish when possible and follow up with a knee for the kill, but do so sparingly and only when you know you can land it, otherwise you're going to be punished hard for your mistake. Also, Grab -> F-Throw -> Knee/D-Air works in certain situations, though not always, but has proved effective in my matches. Use Side-B very sparingly as it's rather slow and easy to get punished. The way I use it is to retreat and then suddenly Side-B towards them. Tends to catch them off guard if they're hellbent on chasing you. Always follow up Side-B with a N-Air, U-Air, or the knee (Which one you choose depends on their damage). Falcon kick is a random, but effective move to use, but only when it's not predictable from a mile away. As I said, use it randomly on an overly aggressive opponent or even as a tech chase in some situations to catch them off guard. Hell, I've even used it to kill going from on-stage to offstage when they are directly horizontal with the ledge and at a high percentage.
Falcon Punch? Never use it unless you are 100% sure you're going to win the match and have a massive lead, or if you want to go for an epic kill off-screen (Highly unrecommended, but funny if you can pull it off, but will result in suicide 99% of the time). Last but not least don't forget to cover your *** with B-Air when an opponent gets behind you or when retreating. That's about all I can think of off the top of my head. There's more advanced techniques but those are really the essentials. Hope this helps, and best of luck
Oh one last tip. If you're offstage and the opponent is offstage with you and below you, you can use Side-B to spike them and bounce off them to grab the ledge. Be careful with this move, however, because if you miss, there's no coming back.