After entering the Citadel, you will have to watch the conversation between Doctor Li and Elder Lyons.
Once the conversation's over, head to the Citadel's Laboratory.
Once you are inside the Citadel's Laboratory, go and talk to Scribe Rothchild. He will give you access to a Vault-Tec Computer to locate the G.E.C.K.
Once you've finished talking to Scribe Rothchild, exit the Laboratory and head to Citadel's Ring A.
After entering the Archives, access the computer and search for Vault 87, that's where the G.E.C.K. is located.
Head back to Scribe Rothchild and let him know that you've found out where's the G.E.C.K. located.
Follow Scribe Rothchild back to the Laboratory.
Once arrived at the Laboratory, he is going to show you where's the Vault 87 located. He is also going to tell you that entering the Vault 87 is impossible due to the high radiation level and suggests you to enter the Vault through Lamplight Cavern.
Then, head to the Little Lamplight.
After arriving at Lamplight Caverns, Mayor MacCready will stop you at the gates. You can convince him to let you in with a Speech check or using the Child at Heart perk.
Once you're inside Little Lamplight, you will have to go through the Murder Pass.
Murder Pass is full of Super Mutants, so you will have to watch out.
After passing through the Murder Pass and reaching Vault 87 door, the quest will end.