Hercules' strength was a product of his divine birth, not rage. And the fact that he killed one guy because he was angry once doesn't make him a barbarian. And if "killing someone while angry once or twice" is the requirement for being a barbarian, almost all of the Greek Gods (and a whole lot of heroes) would be Barbarians. Hercules didn't fuel his fighting style with anger most of the time. Normally he was pretty smart and tactical, quite often solving his problems with his charisma or wit instead of brute force (figuring out how to kill/beat the hydra, tricking Atlas to retrieve the Apples of the Hesperides for him, convincing Hades to let him take his dog on a walk to his cousin's house, etc).
So, Hercules' strength and fighting capabilities didn't come from anger, his one case of "going mad" and killing his family was godly-induced, and he often used his intelligence and charisma to solve his problems, even when it could have been solved with brute strength. If anything, Hercules was a Fighter, not a Barbarian. Battlemaster, probably. Or maybe a Paladin, given his divine connection.