To reach this ending, you have to go through the Untended Graves (an optional area beyond the Consumed King's Garden) to reach an alternate, dark version of Firelink Shrine. Go to where the Irina of Carim usually would be and break an illusory wall to obtain the Eyes of a Fire Keeper.
Give this item to the Firekeeper at normal Firelink Shrine. After defeating the final boss, do not touch the bonfire. Instead, use the summon sign that appears to summon the Firekeeper, resulting in the ending for this achievement.
IMPORTANT! If you kill the Firekeeper after summoning her, you will get a different ending that isn't tied to an achievement.