By doors, I take it you mean main entrances to the palace. In terms of regular doors, there must have been a multitude, too many for poor old Bilbo to count in all his snooping!
The Elvenking's realm was really not so much of a castle, but rather more like an enormous palace carved into a mountain with a "huge cave that ran into the side of a steep slope covered with trees" (155). Bilbo spends much of his time discovering the secrets of the Elvenking's palace.
Although it seems like there is only one main entrance to the palace, which is the Great Gate at the mouth of the cavern, the hobbit finds second way out through a trapdoor in the cellar which accesses the river. Bilbo cleverly stows the dwarves in empty barrels to make their escape down the river to Lake Town.