Falcon is an insane combo character who can aggro all over the stage and wreck you. If you give him the chance, he will pressure you until you give and then combo you to death. He's probably the best in the game at what he does. However, these traits come with some serious glaring weaknesses which some go so far as to argue make him a mediocre character in PM.
First off, he's combo bait. He's a fast faller. A lot of people don't realize this, but falcon is actually about as fast of a faller as Fox. It just doesn't seem that way because fox's fast fall is faster. He's heavier than fox too, so he stays nice and tight with you while you combo him. On top of this, he has some of the ****tiest tech rolls in the game, so tech chasing him is incredibly easy.
Next, his recovery sucks. Falcon's dive is ok in length, and can actually cover a surprising amount of horizontal distance, but it's very predictable and one-dimmensional. The fixed side-b improved his recovery game slightly, but it's still not a great option. Gimping falcon is probably the easiest way to kill him. Combine being combo bait with this and you can get some ridiculously early kills on falcon.
Those two are probably the biggest and most abusable weaknesses falcon has. Other than that, there's the fact that he has no projectiles. This isn't the biggest issue in the world since falcon basically IS a projectile, but it can make dealing with other projectile characters difficult. You could probably be smart about how you dance around the stage while zoning with projectiles. Basically you'll want to shut down his pressure game while zoning and look for openings to combo him. I don't know much about Tink, but I'm sure he has something good to gimp him to death with, and I'm sure he has plenty of combos on fast fallers.