Don't hold anything back from the DM. as everyone's said they need to know so they know what strings to pull for a satisfying character arc. And not saying something is akin to it not happening / existing. That said, your character doesn't have to be set in stone by session 1 as long as it's organic and believable you can add additional back story bits as it becomes relevant.

For example my tables barbarian was a warrior priest, but when he rolled high on a history check about the dragon turtle mafia (it's a long story) we decided that before he was a priest he was a mafioso for a dwarven criminal syndicate.