To begin this quest you must speak with Anthousa, who can now be found at the Leader House in south Korinth. When you speak with her, she will suggest a plan to kill the Monger which involves meeting her at the theatre. You can agree to do this, or alternatively you can decide to kill the Monger in the way that Brasidas proposed; by hunting him in his cave.
Note: Choosing to kill the Monger in his cave will allow you to persuade Lagos, another cultist operating in Elis, to leave the cult without bloodshed later in the game. If you kill the Monger publicly, however, he will refuse to listen.
If you choose to face the Monger at the theatre, simply go to the specified place and wait for him to show up. You can then fight him along with a number of other bandits. Focus on taking out the bandits with a flurry of swift attacks, but be sure to avoid being cornered and attacked from multiple directions. When all the adds are dead, focus on bringing the Monger down.
If you choose to go to the cave, there will be less of a show but more there is more loot to be had. The entrance to the cave is located south west of the Temple of Apollo in Korinth. Enter the cave and follow the path until you reach a statue. A cut scene will then initiate and you will be introduced to the Monger. When the cut scene ends you will have to fight him and four of his thugs, but first you will acknowledge him in the Cult matrix.
When you are ready, return to the game and be prepared to take out the light thugs first. They have weak armour and low health, so a few light attacks or special skills will make short work of them. When you are left with the Monger, evasion is key. Each time you avoid one of his heavy attacks you should counter it with a series of your own. He has no hidden abilities that might take you off guard. When his health is low enough he will drop to the ground. Speak with him to initiate a short cut scene. You will then have to decide whether to kill him in the cave or take him outside to allow all of Korinth see him die.
Note: Be sure to loot the cave for all of its ore.
If you decide to agree with Anthousa and bring the Monger to the theatre at the last minute, you can also decide there if you should kill the Monger or allow Anthousa and her girls to do it. Either way, when the deed is done Anthousa will provide you with some valuable information and you will be able to question her. When the dialogue ends, the quest completes.