I know it's an old question, but I think this is a better, canonical answer.
According to wookipedia, the Clone Wars TV show is still officially canon. There are several episodes in season 2 where Boba is trying to get even for Jango's death (I've watched 20-22, but I think there are some in later seasons I haven't seen yet).
In episode 20, Boba hangs out with a bunch of other clones his (apparent) age (not sure on actual ages), acting like a new recruit so he can attempt to assassinate Mace Windu. At this point, he probably is quite aware he's a clone.
In episode 21, there's an exchange between Anakin and Mace (near the end of the episode, around 18:40 on the Netflix version):
A: Whose helmet is that anyway?
M: It belongs to a bounty hunter I killed on Geonosis by the name of Jango Fett.
A: You mean the clone template?
M: Yes. Strangely enough, he had a son. Or at least a clone he regarded as a son. His name is Boba Fett.
I don't think that confirms anything not already in the prequel movies, but I haven't seen them since they were in theaters so I posted it anyways.
In episode 22, we hear an exchange between Boba and a captured clone soldier (opening scene, right after the intro prologue, 0:45 on Netflix):
B: This isn't what I wanted.
C: You don't have to do this. You're not like them. I can tell.
B: What do you know, old man?
C: I know a good soldier when I see one.
B: I'm no soldier. I'm no clone, not like those two.
It seems pretty explicit from that line he knows both that he's a clone, and that he's different from them in some ways.