Rafiki's character may be one of the most interesting in the movie. He is a wise old Baboon, sort of a High Priest or Shaman of the Pride Lands. He cares very deeply for the welfare of the Royal line, and the Pride Lands.
He can have a very awesome presence. He can also act like a madman to get his point across to people who wouldn't listen otherwise. His crazy act is just that - an act. Everyone recognises him for his wisdom in the end.
On another note - saying he is a baboon is only partly true. He's sort of a cross between a baboon and a mandril, but we won't tell if you won't.