This achievement can be obtained by pkaying the main storyline within the Taken King DLC.
The following are the missions that encompass the Taken King Quest. You will need to be running back and forth between the three class trainers, and Ikora.
The Coming War
Enemy of my Enemy
Lost to Light
The Promethean Code
Achievement Glitching for you? Log out of Destiny, and reboot. This should unglitch your achievement.