Hrun the Barbarian is more than just a barbarian. He's roughly every barbarian ever rolled up into a hilarious caricature of the brawn-over-brain mentality: He-Man, Conan the Barbarian, Thrud the Barbarian, and just about any Dungeons and Dragons character with a rocking fortification save throw.
We can see this from the minute Hrun walks onto the scene with his "wide chest," " neck like a tree trunk," and "small head under its wild thatch of black hair looking like a tomato on a coffin" (2.6.9). He's strong, fearless, and can take on two dragons and their riders with his bare hands. He also collects loot and damsels with the zeal of the most avid collectors. In other words, he belongs in a Frank Frazetta painting.
In traditional sword and sorcery fantasy, the barbarian is strong, but he has intelligence to match his muscular prowess. At least, he's smart enough to out-think and outmatch all those dastardly sorcerers and booby-trapped crypts.
And while the heroic barbarian certainly has a love of loot and adventure above all else, he also has a soft spot in his heart for those kindly folk who don't have his muscles or skills in combat. He will often help them defend their village or rescue their daughter or do whatever needs doing, if for a price.
Hrun, yeah, not so much. He's super strong, but his problem solving process pretty much begins with punch it and ends with hack it with a sword. To be fair, he's "practically an academic by Hub [barbarian] standards in that he [can] think without moving his lips" (1.8.33). That said, he retains little interest in others and is a total egomaniac—so much so that Rincewind is able to pay him for his protective services by offering pictures of the barbarian "striking a heroic pose" (2.11.3) over his slain foes.
In a wonderful twist, Hrun's also grown bored of adventuring, the daily grind having worn on him over the years. He's decided that his life to date has been "little more than a reputation and a network of scars" (3.15.85), leading him to quit adventuring to become the king of Wyrmberg.