This happens on your first visit to Wall Market after helping Aerith escape the church. For clarity, the three areas are Lower Wall Market, where you first come in, Upper Wall Market, and the Honey Bee Inn, reached from the bottom right of Lower Wall Market. You'll need about 300 gil to do this optimally, and while you should have enough coming in, there's an equipment store that can eat your money, so do this first.
Talk to the people outside the Honey Bee Inn to learn Tifa went to Don Corneo, then go back to Upper Wall Market and talk to the guard to start. Go back to Lower Wall Market and go to the dress shop, which has clothes outside, and talk to the clerk. Back to Upper Wall Market, go to the bar which has an IN doormat and talk to the person on the right side. When prompted, choose something that feels soft and something that shimmers to get the best dress. Go back to dress shop and try it on to be prompted to get a wig.
Back in Upper Wall Market, go to the gym, which has a neon green sign and two people talking outside to the left. Talk to the woman there to get into a minigame; as long as you beat Big Bro, who typically does 15 squats, you will get the best wig.
Next, in Lower Wall Market, go to the shop with neon blue signs across from the dress shop, though you'll have to go down and around to get there. He'll ask you to check out the vending machine at the inn; the inn has a person standing right outside the door. Stay for 10 gil, and when prompted, buy the 200 gil item. Take it back to the guy to receive a Diamond Tiara.
The next stop is the diner, which is the shop you had to run past between the dress shop and the tiara guy. Have a seat, choose any food, and choose "It was all right" to get the Pharmacy Coupon. Take this to the shop under the diner with a cat statue, and trade the coupon for Digestive. Take that back to the bar in Upper Wall Market, go to the top, open the bathroom, and give the woman the Digestive to receive Sexy Cologne.
Make your way towards the Honey Bee Inn, and talk to the person in the white shirt in Lower Wall Market before the transition to receive the Member's Card, which will get you into the Honey Bee Inn. Once inside, choose either of the two rooms on the left. It doesn't matter which, as this item isn't the important one; go to the room at the top to enter the changing room, and talk to the girl on the left to put makeup on Cloud.
From here, SAVE. There's still a low chance that Don Corneo will pick someone else, at which point, just reload the save and try again. Go back to the dress shop, put everything on, and go back to the guard to be let in. After talking with Tifa, you'll meet with Don Corneo, and after a few lines, he should select you, giving the achievement by the time you're in his room.