Unfortunately, while there are quite a few notes about his game-playing around the 'net (he's also a big Guitar Hero fan, so Activision-Blizzard will probably appreciate that), we haven't seen his characters mentioned anywhere, so we have no idea what class or race he plays. For some reason, Mage jumps to mind as a complete guess, and so does Horde, but we'll probably have to ask Kluwe himself to find out.
Update: Internet detectives in the comments below say he's a Troll Rogue on Kil'jaeden. And that he's already hit 80 (isn't he supposed to be practicing for the playoffs?).
Update2: Kluwe got in touch with us and confirmed that yes, he's an 80 Rogue (so I'm 50/50 on the random guessing). He also says that he hasn't had too much time to play lately, since he and his wife just had a baby girl. And he says that if you really want to be amused at how he spends his time, check his achievements. Yes, he's a Merrymaker.