Complete the quest picking up the trail.
You get the quest from Scribe Rothchild. he will reveal the location of the Citadel's Vault-Tec terminal. Once you've looked at the terminal Rothchild tells you the location of vault 87 and suggest you go to Lamplight Caverns.
Once you get to Lamplight you will be stopped at the gates. Once inside you have 2 options.
1.Passing through Super Mutant infested Murder Pass; if you chose this option, you must first speak with Mayor MacCready or Princess to get the gate to Murder Pass open.
2.Asking Mayor MacCready about alternate paths will reveal a locked back door to Vault 87. You must ask Joseph to restore power to the computer terminal before you can use it. The door and terminal (average difficulty) are located in the Great Chamber.