Talk to Astrid who will pass you along to Festus Krex. He will tell you to kill the Gourmet so you can take his place. Track down Anton Virane to get the Gourmet's real name, then kill him. You will find him in the Temple Kitchen at Markarth. He will give you the Gourmet's name.
Now head to the inn and kill Balagog. He will be in the cellar which you can simply walk into. Kill him in his sleep and drag his body to the ladder at the end of the hall to receive your bonus. Turn in the quest.