You need to kill 3 Queen Greems for this achievement. It's easy to keep track of because you get the objective to kill the Queens through a side-quest from a man known as the Greem Exterminator. He is located shortly west of the Mayor in Pluckmuckel and looks like the person in the achievement picture. He'll tell you that his father needs to be avenged and that you'll need to kill the Queens for his honor to be returned. He'll give you a whistle and the location and you must do the dirty work.
Each Queen Greem is located at a mating stone in a certain area. Walk up to the stone and use the whistle you get for each quest with the stone. Walk up to the stone and press to go to your inventory. Select the whistle and the Queen will appear. Once you kill her walk over her dead body to recover the Queen Greem Sack and return it to the Greem Exterminator. Do this three times and you'll net your achievement.
The first Queen Greem (Lvl 15) is located at the Greem Lake Mating Stone. The best way to get there is to use the Outhouse teleporters, travel to the Slimey Outhouse and travel southeast.
The Second Queen Greem (Lvl 18) is located at the Haunted Forest Mating Stone in Zombie Greem Camp. The best way to there is to use the Outhouse teleporters, travel to the Haunted Lake Outhouse and travel north.
The Third Queen Greem (Lvl 20) is located at the Farmhouse Mating Stone. The best way to get there is to use the Outhouse teleporters, travel to the Wretched Outhouse and travel south towards Greem Camp.
Once you kill the last Queen the achievement will unlock. You don't have to turn the quest in to get the achievement, but I suggest turning it in anyway for the experience points.