Both Kirby and Charizard are relatively unused in competitive play making this particular match-up rarely seen. I don't run into many Charizard players in tournaments, and the ones that use him generally are not high level players. Despite this, I figure I may as well give some input.
In general both Charizard and Kirby will want to close the distance and bring the fight up-close since neither character has a particularly strong long-distance game. Kirby will generally want to get in closer while Charizard can space a little bit farther out with longer-ranged pokes or possible Flamethrower damage. However, lower-skilled Charizard players will generally Flare Blitz at any time and any distance.
Charizard's spacing isn't safe enough to keep Kirby out, and when Kirby gets in he can deal a good amount of damage with his combo game. Kirby's footsies against Charizard is safer in speed, but not in range, which means Kirby will try to get in close and poke at Charizard. Kirby will likely mix in a lot of Shield and Dodge with his pokes. The reason is because Charizard can eat right through Kirby's attacks such as with Rock Smash, Flamethrower, or brute force with F-smash. Kirby's pokes combined with his defenses work to bait something giving Kirby an opening to move in and attack.
Grabs are a big part of this match-up for both Charizard and Kirby. Charizard gets good damage in with his Throws in general, but mostly with D-throw combos. Since Kirby is so lightweight he can't afford to take many hits and Charizard's grab is a great way to beat out Shield and wrack up some damage. Grabbing is a great option for Kirby as well, and most of the time if Kirby has an opportunity to punish it will be with a Grab. Kirby will generally use F-throw to set up his combos at low to mid percents, so DI away from it.
Up-Throw is the usual go-to for a KO throw with Charizard, and while it's not nearly as strong of an option on Kirby's side of things Kirby can also go for U-throw KOs. Due to the weight differences between Charizard and Kirby as well as the fact that Charizard's U-throw is stronger, Charizard can get KO's with U-throw at much lower percents than Kirby can. Use platforms to your advantage and keep in mind how low the ceiling is on whatever stage you are playing on. For reference: Kirby's U-throw KO's Charizard at 154% (no Rage, no DI, Final Destination).
Most of this battle really comes down to making the best choice and not getting caught by a bait. Overall I feel like Kirby has the upper hand just because he has safer options and thus a stronger bait & punish game. Again, I'm not too adamant on this simply because of the lack of high level interaction these two characters have as of 1.1.3. I'd love to hear things from the Charizard perspective, as well as what other Kirby players think.