This Quest is only available for Hunters and unlocks your third sub-class, the Nightstalker. Complete the very first mission of The Taken King's storyline, and Cayde-44 will have the quest available.
The quest is a two-part mission that can only be played solo, and takes place in two different areas. Completing the first area will automatically send you to the second, with no ability to change destination, but the quest is short with few enemies.
Once you complete the second area of the quest, you will have the Nightstalker sub-class, but you must still return to the tower and turn in the quest to Cayde-44, and only then will the achievement pop. Enjoy your new subclass and gamerscore!