Once you are in the room he came from, you are home free. Other then that, you can read any La Fantomas that you have, make sure to keep the light on your pip-boy off, or try wearing the assassin's suit, which is located in the Medical district of the Villa. If you are having problems staying undetected, there is a gun case in the same room as the terminal that has a chance of having up to six stealth boys in it. Once he reaches that area, start moving (still crouching) and remain undetected until you reach the area that he entered through. Once he comes down, go into crouch mode (if you aren't already) and wait for him to reach the area where the computer and vault door are. (A little extra help, the lockpick and sneak challenges can be activated by telling Elijah that you will take the treasure for yourself) Once you can move again after he says he's coming down himself, run out the vault door and up the steps to the left, but do not go through the doorway to where you entered from, just wait in the dark there where you can still see the vault door and the bubble around it. A repair, science, or explosives of 75 will do, and if not, a lockpick or sneak of 75 will make him come down after you.
Before you actually trap Elijah in the vault, you have to convince him to come down.
