How do you join the brotherhood of steel outcasts?

The Brotherhood Outcasts are located in the northwest of the Capital Wasteland, in a facility north of the SatCom Array NN-03d, north of Germantown. Once you have reached this location, you can join the Outcasts by speaking with Protector McGraw, who will inform that the Outcasts are on the lookout for new recruits. To become a recruit you will need to complete the quest The Outcast Outpost. Following the quest you'll be accepted as a recruit, which also enables you to trade with Outcast vendors.

To become a full member, you will need to complete the quest Into the Great Wide Open, given by Elder Simms. It requires going to the Adams Air Force Base crash site and retrieving a flight recorder from the wreckage of a crashed vertibird, which contains the location of a Brotherhood base in Chicago, The Outcast remnants will leave their compound near Satcom Array NN-03d with you following them to the crash site, where you will need to help them fight the Talon Company forces at the site, then enter the plane for the recorder. Once the flight data recorder has been retrieved, leave the aircraft wreckage and return to the Outcasts' compound. When you return to Elder Simms, he will congratulate you on successfully completing your tasks and becoming a fully-fledged member of the Outcast remnants, and will reward you with a number of items, including Outcast armor and a Brotherhood Outcasts power armor.