I rarely fight against Captain Falcon outside of Falcon dittos. However, the same advice applies in any matchup against the Captain.
As has been discussed, CF's combo options are very deadly once he gets on a roll. He has one of the best aerial games in Smash when he's below you, so he'll generally go for a launcher or throw followed by a chain of aerial attacks.
He's the kind of character where even one small opening or mistake can lead to a huge string of attacks in seemingly no time at all.
If you find yourself stuck in one of his aerial combos, don't try to respond with attacks of your own since these will either trade or get outprioritised. Go for the airdodge, get to ground and try to get the stage to a position that favours you, or at least try to reset the stage.
Clever Falcons will start to read your airdodge tactics after a while and may use this to land his F-air, the Knee (a devastating killer move) or more U-airs. Mix things up to ensure he doesn't get the upper hand on you.
However, despite his excellent combo potential, Falcon has several weaknesses. As a Falcon main, these are the areas that bother me the most:
1) Projectiles, and projectile spam. He lacks a projectile of his own and is particularly vulnerable in For Glory with its flat stages, so use those as you see fit. However, he can cancel many projectiles with his jab.
2) Vulnerability in the air when he's above you. His U-air is legendary but he doesn't possess anything similarly fast for dealing with foes below him. If you launch him, take advantage of that as much as possible and try not to let him dodge you on his way back down.
3) Gimpable recovery. This can be particularly useful if you have a projectile, but it's viable to mess with his recovery even if you don't. Just don't underestimate his up special either since its grab effect goes straight through many attacks.
4) Inability to respond well when under attack. Get him in a good combo and his options are limited, other than to airdodge and evade where possible. Use this to your advantage.
5) Ness, and Wario. These two characters can prove bothersome for Falcon, particularly Ness. However, this really comes down to the skill of the player.
Again, give Falcon the opening and he'll destroy you in the air. Try to dodge him, read his play style and keep him in a position where you have the advantage as much as possible. And, of course, show him your moves.