After you turn in a quest in Fort Dawnguard, it might say "Speak to Gunmar for an assignment". This objective should show up under the Miscellaneous section in your quest log. If you set this objective as active, it will lead you to Gunmar who should be on the first level of the fort.

Gunmar will give you the following items the first time you talk to him for an assignment:

Dawnguard Heavy Armor
Dawnguard Boots
Dawnguard Gauntlets
Dawnguard Helmet

You can also buy weapons from him, including a Dwarven War Axe, Elven War Axe, an Orcish Bow, as well as armor including Elven Armor, Elven Boots, and even an Elven Helmet.

Gunmar can also train you in Smithing.

The coolest thing you can buy from Gunmar though, is an Armored Troll (for 500 gold) that will follow you around and help you fight. This is one of the perks of being a Dawnguard rather than a Vampire Lord.

To get the side quest, ask Gunmar what you can do to help. One of the quests you get is:

Hide and Seek
Gunmar has sent me to Morthal. I am to find and kill the vampire masquerading as a traveling pilgrim. I am to do so discreetly as the law is unaware of the thread and will assume I have murdered a citizen.

Objective: Discreetly kill Traveling Pilgrim

NOTE: Location and other details about the quest will vary, as it is random. For example, instead of a traveling pilgrim at Morthal it could be a Visiting Advisor at Dragon Bridge, a Traveling Bard in Solitude, or a Traveling Merchant in Riften or Rorikstead. Regardless, the actions you will need to take are the same, so follow this walkthrough but use your target's location instead, if different.

Gunmar tells you he has located the whereabouts of a vampire masquerading as a common citizen. He wants you to track him down, and dispose of him. Tread lightly, The law is unaware of the danger and will treat you as a criminal if you are caught.

If you ask Gunmar about informing the Jar's men of the threat, Gunmar says that the vampire can cloud the minds of those he preys upon. It won't be easy to convince anyone he's other than he appears. The law will require proof that we can't present. And Jarls disapprove of bloodshed they haven't called for themselves. Best not to damage our reputation. Let your actions be as discrete as possible.

If you ask Gunmar how you can avoid trouble with the law, he tells you that you must be clever. Find a way not to be caught or accused of murder. You'll figure something out. We must be vigilant and self sufficient if we are to defeat these monsters.

You can also ask what the vampire is doing. Gunmar tells you that vampires enjoy stalking their prey. Passing among us undetected emboldens them. Makes them feel powerful, untouchable. But we'll put a swift end to that.

Exit Fort Dawnguard and travel to Morthal. Once there, locate your target. Your mark will more often than not be in an inconvenient, non-discreet location such as inside a tavern, or sitting outside a building near a group of guards.

How to kill discreetly?

You have several options:

Talking to the target
You can try taking to the target and it will display options such as:
"To which shrine are you headed, pilgrim?"
"You're no pilgrim"
"Come with me, pilgrim"

Tell him to come with you. It then presents you with another choice:
"I'll take you to the monster you're hunting" (PERSUADE)
"Follow me, or you'll regret it" (INTIMIDATE)

If your speech skill is high enough, you can make the pilgrim follow you to a hidden location, where you can then kill him discreetly. Try persuading first, and if that does not work, try the intimidation. Once you have gotten the target to a secluded location, kill him.

Transform into Vampire Lord or Werewolf
If you can transform into a Vampire Lord or a Werewolf, you can kill him without being recognized, although this still attracts attention.

Wait until he is in a secluded location
You can wait until the traveler walks to some hidden location such as an alley or corner where noone can see. Kill him then

Use your thieve guild or thane authority
If you are the Thane of the city, or a member of the thieves guild, you can kill him then ask the guards to forgive the crime

Accept the Bounty
If you are not of the patient type, just kill him on the spot. You will get arrested and have to pay a high bounty, but the quest will still be completed.

Return to Gunmar and report your success, to complete this quest.