Flamethrower catches Fox's up b and side b, even if Flamethrower is stale, so if Fox tries to recover low, it's game. Fox can mix up his recovery, but Charizard will still have stage control and then he can mix up his edge guarding. Flamethrower shuts down most of Fox's options, and recovering high only gets him punished. I've played a lot of good Fox players and they all fall to Flamethrower, and it's not only Fox, either. Charizard's edge guarding game just happens to be one of the best, if not the best, and many characters suffer from it. He can wall out most characters with short-hopped forward airs, too.
Fox does have the tools to pressure Charizard, and Charizard does get combo'd by him, but when it comes down to getting the KO, Fox needs to get a hard read because he lacks a reliable way to take down Charizard. The lack of kill throw is also detrimental. That's when Charizard has the upper hand because Charizard is all about prediction and punishing, and thanks to the high percentage that he'll have from Fox's comboing, he can KO early because of rage. Charizard also has one of the best kill throws in the game now so Fox can't be pressuring his shield unless he wants to get grabbed and KO'd/thrown off the stage. Fox becomes incredibly predictable when closing in on the kill, and Charizard gives Fox a beating if the Fox player isn't playing flawlessly. And no one plays flawlessly.
Fox has set-ups and the combo game. Charizard has the edge guarding and the punish game. Fox will be ahead most of the match, but Charizard can quickly turn that around if the Fox gets cocky or hiccups. Rock Smash shuts down Fox's KO options, so even if a Fox player thinks he has a read, he probably doesn't because the Charizard will counter the read with his own.
The match-up is even in my books because it all comes down to who gets the punish. They both screw each other up bad. It's even better for Charizard if custom moves are put into the equation. Dragon Rush and Rock Hurl give Fox a very hard time.
The same thing can be said for the Captain Falcon and Sheik match-up.
Alternatively, and being heavy Charizard suffers from this greatly, you could use a sour spotted Fair to spike him if you're good at it.
I wouldn't recommend that since Flare Blitz has heavy armor, and Fly has super armor. The Fox will only get stage-spiked.