The Barbershop room unlocks at 50 Vault Dwellers, and costs 10,000 Caps to purchase. It takes up 2 blocks, and is a Charisma based workroom.
Changing a dweller's appearance will cost time and Caps. Switching a dweller's hair color will cost 100 caps each time, which seems to be the optimal route for getting the achievement.
The upgrade costs 50,000 caps and unlocks even more customization.
EDIT: It appears that higher charisma stats decreases the time to customize a dweller, so the (sexy) sleepwear or other Charisma boosting outfits are good for the ones that are trying to do this in the least amount of time.
EDIT 2: A lot of people are experiencing a bug that locks their achievement progress at a certain percent. According to UnfurledEmu75, a possible workaround would be to create a new vault and work your way back up to 50 Dwellers to customize 10 new ones. Unfortunate, but there seem to be a handful of glitchy achievements in this title, so you can work on this if you have another glitched achievement by the end of your completion. Just save it for later.