Why Did The Mad Hatter Have A Split Personality?

Yes, there was a backstory and it was actually introduced in the film:
the whole hatter clan died which Tarrant (Hatter) was in, so he lost everyone from his "family".
Another reason for his "craziness" was probably that the red queen had taken over wonderland and Alice was not there to help them.
And the last point is, which was not mentioned in the film, that hatter(s) back then got poisoned from the mercury which was in the glue for the hats- and this mercury made him (them) mad. That is where the term "Mad as a hatter" came from. Hope I could help.

Also, Tarrant having a Split personality was Johnny Depp`s idea, as we can see in the official interview of the movie. He made it up on the mercury problem mencionated before by hattersmadgirl

Mad hatter is just mad, or Insane AND he drinks to much tea with his insane friends.