To get the duke dementia, shivering isles achievement, do i have to go through the gate of mania or the gate of dementia?

It doesn’t matter which gate you choose. At some point during the game, you will have to choose between lighting the Fire of Dementia or the Fire of Mania, in the Church in New Sheoth. Choose Dementia if you want to continue on the road to becoming the Duke.

If its just for the achievement, save right before you get the choice and do both then actually use the one you preferred.