Directly after The Waters of Life ends, this quest begins. You will be instructed to find the location of a G.E.C.K. Scribe Rothchild may be able to help. Find him in the lab and he will point you to a Vault-Tec computer. Check the Vault 87 item list to see it has a G.E.C.K. Return to Rothchild to learn of the vault's location. Unfortunately you will have to go through the Lamplight Caverns because of radiation levels outside the vault.
When you arrive at Little Lamplight you will be stopped by a child, Mayor MacCready. You have three ways of getting into the city. With a high speech skill you can talk your way in. With the Child at Heart perk you can also talk your way in. Without one of those, you will be forced to do the Rescue from Paradise quest. If you have already done that you can get in no problem, if you haven't you will need to do so now.
Now that you're in the city, you need to access Vault 87. You can either use speech on MacCready or Princess to gain entrance to Murder Pass which will be filled with enemies, or you can use speech on MacCready to learn about the door that won't open. Once you learn of that, speak with Joseph to restore power to the terminal for the door. Hack the terminal for entrance to the vault.
See Picking Up the Trail for more specific info.