You will start the quest being held in a force field after being captured in the previous quest. Colonel Autumn will ask you to give him the access code to the purifier. Giving him the true code will get you killed, the best choice is to play it dumb until President Eden will ask him to leave and release you from the force field and ask you to meet him so you can speak with him. After being released, don't forget to take all of your stuff back from the Locker.
After going through the level 2 and 3 and finally reaching the control room, go upstairs and speak with President Eden.
He will ask you to take the modified FEV virus and insert it into the purifier's console at the Jefferson's Memorial, which will result in a bad ending. You will still have to take it whether or not you're planing on using it, just so you can exit the Control Room
You can also convince President Eden to self-destruct by passing a few Speech checks.
After talking to President Eden and taking the FEV virus, escape the Enclave base.
Once you've escaped the Enclave base, return to the Citadel.
After speaking to Elder Lyons and letting him know about the G.E.C.K. the quest will end, leading to the last one.