You have to feed your dragon peppers. Plus I think there's a multiplier in the config file.

I did a lot of testing. The end results were largely inconclusive, but i do believe leveling up melee damage directly affects the damage of your firebreath. however there are many other factors as well, such as mate boost and damage resistance. the most important of these factors is range, being closer to your target you do more damage, that's all i got.