Sort of a long post, but this character is interesting and very tragic if you take the time to think about it and consume depth from the novels.
He had an exceptionally shitty upbringing because of Grom, which founded numerous mental issues. After being recruited into the Horde, he spends his time acting tough, trying to impress others and that he is worthy of dad's legacy, like what he did with Alliance sailors in front of Cairne. But other than the Orcs serving under him, no one really likes or respects him. Thrall then slaps Warchief title onto him hastily despite his vocal doubts.
Garrosh is then repeatedly antagonized by every other race and their faction leader in the Horde. With Vol'jin and Baine, he repeatedly tries to impress them and win their favor but is coldly rejected every time just because of previous tensions (unintentionally killing the latter's daddy), and Sylvanas and the Blood Elves couldn't care less. But the Orcs of Orgrimmar love him for his war hero status and bringing prosperity to them where Thrall didn't.
So being a mentally fragile guy that desperately wants approval but keeps getting rejected by all but Orgrimmar, it's not really surprising he would be corrupted by and so easily in favor of the Blackrock Orcs and Malkorok that only ever say yes to him and worship him. You can tell how fucked up Garrosh became in the head mostly when he started associating with these Orcs, especially Malkorok who pretty much became his best friend.
You can just see how fucked up Garrosh is mentally in War Crimes and through his interactions with Anduin, he speaks the way he does and acts in WoD because he feels like he has nothing left but to go on the path he's chosen by now since Azeroth obviously wants him dead.
Overall, I think as an example of a mentally ill character, Garrosh was done well and it pisses me off Thrall denies any responsibility and most people see him as a mustache twirling villain.