Joining the Thieves Guild can be accomplished in a number of ways. The first is to simply steal some items by breaking into a house or pick pocketing someone. Then allow the guards to catch you and throw you into jail for a bit. Once you’re released, a representative of the Thieves Guild will approach you. Alternatively, you can also read one of the wanted posters for the Grey Fox posted around the Imperial City. Doing so will add the “Grey Fox” conversation topic with some NPCs. Find a beggar in the Imperial City and ask them about the Grey Fox. If their disposition is high enough, they will ask you if you are looking for him. If you say that you want to join him, they will tell you to go to the Garden of Dareloth in the Waterfront District of the Imperial City at midnight.
Armand Christophe, a representative of the Thieves Guild, will meet you and two other potential recruits. The test for admission is race to which of the three of you can steal the diary of Amantius Allectus and return it to Armand first. To start, talk to the beggar Puny Ancus and bribe him to tell you the location of the house. He will tell you that it is in the Temple District. The house is actually right next to the door between the Temple District and the Imperial City Arboretum. Break into the house and open the desk on the right side of the room to take the diary. Return to Armand in the Garden of Dareloth, give him the Diary, and he will admit you to the Thieves Guild. The achievement will unlock as soon as the initial conversation with Armand is finished.