Talk to Feran Sadri or any other vampire in Castle Volkihar. They send you to Fura Bloodmouth for an assignment. Get your assignment from Fura. She wants you to kill a Dawnguard spy. But don't draw attention. Some kills are for fun, some are for profit. And others are for sending messages. This one is a message. With any luck, he'll bleed to death before anyone notices. A cold corpse will say what we want better than a warm one.

You can ask Fura Bloodmouth what she means by "don't draw attention". She clarifies that she does not want you to make a big scene, and you should definitely not get caught. The corpse is to be the message, not the attack.

This starts the quest:

The Hunt
Fura Bloodmouth has sent me to Ivarstead. I am to find and kill the Dawnguard member masquerading as a Traveling Bard. I am to do so discreetly.

Objectives
(Optional) Make it a discreet kill
Kill Traveling Bard

Note: The NPC you have to kill and the location are chosen at random and will be different each time. For example instead of killing the Traveling Bard from Ivarstead, you might have to kill a Traveling Bard from Riften, a Traveling Pilgrim in Riverwood, or a Traveling Merchant from Rorikstead.

Travel to Ivarstead (or any location you were assigned) and find your mark. Wait for him or her to be at a place hidden from sight, then sneak from behind and assassinate him. You can also attempt to talk to him and persuade to come with you, then lure to a hidden area and kill him there. If you get spotted, bounty will be added and you will fail the optional objective, but still complete the main one.

Return to Fura Bloodmoth at Castle Volkihar and tell her you have killed the Dawnguard masquerading as a traveling bard. You get an enchanted weapon or armor piece as a reward.

This completes the side quest, "The Hunt".