Instructions
Travel to the south entrance of Varrock during the Shield of Arrav quest. Start a conversation with Charlie the Tramp while there. He will show you where the Black Arm Gang's hideout is. Follow his directions and enter the gang's hideout. In there a woman named Katrine will be present.
Speak to Katrine. Refuse to reveal your source as to how you heard about the gang and tell her that you're trustworthy. Katrine will give you a task to complete before you can join the Black Arm Gang.
Get a friend to join the Phoenix Gang. Have your friend give you the key to the Phoenix Gang's weapon store. Travel to the Phoenix Gang's hideout and open the locked door with this key. Climb up the ladder present.
Kill the Weaponsmaster in the Phoenix Gang's weapon store. Take two Phoenix crossbows on the way out.
Travel back to the Black Arm Gang's hideout. Hand her the two Phoenix crossbows and she will make you an official member of the Black Arm Gang.