The Endless Nightmare is Act 10, and must be unlocked by first completing the game on any difficulty. The difficulty setting for Act 10 is "Torture" and cannot be changed. The goal is to complete each stage in the time allotted, without dying. You only get one life to complete each scene, so if you die, you must start over again.
To get this achievement, you must complete all three scenes in Act 10, which are selectable from the Scene Select menu. See the three achievements for obtaining an A rank on each scene for more.