Fallout Shelter

You unlock the Barbershop when you get 50 dwellers in your vault. In order to establish some base line stats, here is the wiki regarding the barbershop. You can see that it takes about 10,000 caps to build the initial barbershop, which is really all you need for this achievement. So you need to alter the appearance of 10 dwellers in order for this to unlock.

The main stat in the barbershop is charisma, which means that people with higher charisma will do this much faster then those who have lower. If you have any high charisma dwellers, you should send them first while sending in those with lower later. Or, you could use gear and pets, assuming you have any, to buff the low charisma dwellers to do this faster. So in order to get this achievement as quickly as possible, here are the two aforementioned methods you can use:

High Charisma:
1.) Build the barbershop
2.) Send high charisma dweller to the barbershop
3.) Change hair color or hairstyle (usually the hairstyle is the cheapest at about 200 caps, depending on which style you're going from->to)
4.) Usually this will take about 5-10 minutes so change when finished

Low Charisma - With Gear:
Small note - On this wiki with all of the outfits, you will notice that the ones with the highest gear output, on the common level, will be the Nightwear (+3), Junior Officer Uniform (+1), Initiate Robe, BOS Uniform (+1), Mechanic Jumpsuit (+1), Vault Socialite (+1) and the Business Suit (+4). Of these choices, the Nightwear and Business Suit are the highest. But, in my own experience, the Nightwear and Initiate Robe are the most common. So if you haven't yet built an Outfit Workshop, these may be your most available.
1.) Equip a dweller with roughly 5 or more Charisma with the best Charisma gear you have.
2.) Send them to the barbershop
3.) Change hair color or hairstyle (again, usually the hairstyle is cheapest at about 200 caps depending on which style your going from and to)
4.) Wait out the, slightly extended time due to lower than 10 charisma, 7-12 minutes for this to finish before sending in the next. The time will differ depending on your level of charisma.

Here is a post from TunderProsum on reddit that goes VERY in depth to not only the math and more eccentric processes of the game, but will aid you in further going about how to level your characters and build things in an effective manor. (Don't miss the Google Sheet he/she linked to. It's a page MANY have contributed to and is very useful)

I do hope this guide has helped! If it has, please do give a thumbs up so I know. Otherwise, if you believe there is something I have missed or should elaborate more on, feel free to give me a thumbs down and explain what I should be including so I can earn that thumbs up! Good luck fellas/lassies.