Feed thy noble queen 3 times during one visit to her quarters

Once you complete the main storyline and the 'Queen' has returned to her sanctuary, fast travel to Flamerock Refuge. Once you arrive, turn to face north and run up the small hill past Moxxi's on the left and the clock tower on the right. Dead ahead kinda northeast-ish you should see the queen in her sanctuary.
Talk to her to initiate a quest and then again to complete the quest. After this you should be able to talk to her again and then feed her by pressing Y. Repeat another couple of times.
It costs 5 Eridium to feed her so 15 are required to net the achievement.