Unfortunately, Captain Falcon has inherent weaknesses that give him a hard time to compete with top tiers in the tournament scene nowadays. To compensate, Falcon players must utilize every tool available to them if they wish to beat those nasty tier abusers. Here are some basic combos that will win you games in clutch situations:
Jab -> Left Hook (The Dazzler): This is a solid opening move. It will catch your opponent off guard, and, if you're lucky or skilled enough, may even nab you a kill. However, you'll have to watch out for opponents with good reflexes: they may recover out of the hitstun of the jab and dodge the Left Hook.
Shin Kick -> Knee -> Leg Sweep -> Body Slam (Shin Slammer): This is also a good opening move; however, you have to be aware that it takes awhile to start up, and will not succeed if the first hit fails to connect. You have to mix up your approach. This requires flawless technical skill. However, if done properly in the air, it could cause upwards of 30% damage.
Take out your gun -> shoot it (Falcon FIRE!): If either of the 2 combos above fail, I highly suggest this technique. It's fast, reliable, and usually powerful enough to take out the opponent without need of edgeguarding.
Apply these combos appropriately and I can guarantee that you will never lose a match.
Note: These newfound skills may drive your opponents away due to your amazingness. This is a side-effect you will just have to accept.