This achievement is obtained in the last mission. You fight the final boss here. To kill him, just shoot. He has two phases before he dies. Simply shooting him with powerful weapons will work. Use cover to avoid his BFG shot. If the BFG shot can "see" you, you will get hurt, or die if you're close enough.
After you kill him, go back and use one of the elevators you took to get to the boss. Then look around for escape pods. One of their doors will be open. Go inside one and press the button to leave.
Note: The achievements for this game only unlock in Single Player, and on any difficulty.