New primary objective: Kill Anoriath
Note! This quest is optional and a chance to get it will show up during completing the quest The Silence Has Been Broken.
In order to unlock this quest, you have to ask Nazir for new contracts during conversation with him. It is worth noting, that this contract and others will be unlocked only if you have completed all previous side contracts. Leave the Dark Brotherhood's Sanctuary and open the world map. Anoriath lives in Whiterun (screen above), so travel there now.
Anoriath spends most of the time running the market store at the marketplace in Whiterun (screen above) or walking down the streets. I do not recommend attacking him then, unless your character is able to sneak well and attack from a distance. Otherwise you will be immediately attacked by local guards and even if you surrender, you will have to pay a fine or go to jail.
You could wait until Anoriath leaves the town and goes hunting, but unfortunately he rarely decides to do this. So it is best to wait until night, because then Anoriath should go to sleep to the Drunken Huntsman inn (screen above).
Enter the inn and find the hunter in one of the side rooms. A there are other guests inside the inn during the night, do not forget to close the door. After that, carefully approach sleeping Anoriath (screen above). Get rid of him in any way, stabbing him with a dagger for example or shooting an arrow in his head. After successful action carefully open the door, leave the room and close the door again.
New primary objective: Report back to Nazir
You can now return to the Dark Brotherhood's Sanctuary and report Nazir what you've done. If you're going to complete other contracts received from him, you should return to the assassins' hideout only after completing them all. Nazir will reward you with gold.