Once you're inside the Sierra Madre Vault during the quest Heist of the Centuries and you have made Elijah come down to the vault, wait inside for him to stand in front of the door or when he comes from behind the shield to kill you, then is your chance to kill him. The achievement will appear after he is dead.
The achievement picture is similar to the achievement picture of Arizona Killer, except Elijah is under Vault Boy's foot and the background is of the Vault as well as Father Elijah not holding an NCR flag.
Elijah will only come down if you pass the speech checks - see Heist of the Centuries for the complete walkthrough.
This achievement can only be achieved in place of Safety Deposit Box. They cannot both be achieved at once.