To start this quest, you need to first complete a few of Sorine's and Gunmar's mission. Do other Dawnguard sidequests until this becomes available. If you have not done enough Dawnguard Radiant quests to trigger this, than Isran will just tell you to go speak to Gunmar.

Ask Isran, in Fort Dawnguard, what you can do to help. He tells you that the Jarl's in danger. A vampire plays as a visiting "advisor" to the court. Destroy it before it's able to complete its foul business. But be warned, the Jarl's guards are unaware of the threat, and will treat violence against this "advisor" as a crime. Convince the Jarl of the threat and he'll let you destroy the creature. If not, than do your best not to get caught. Quickly now! The longer you delay, the more danger the Jarl is in.

This begins the quest:

A Jarl's Justice
Isran has sent me to Dawnstar to warn Skald of a vampire masquerading as a visiting advisor to his court. I am to destroy the beast. If I fail to secure Skald's permission first, the guards will consider it  a murder.

Warn Skald about the vampire threat.
Find evidence linked to the vampire

NOTE: The Jarl and City you get are Random.

Talk to the Jarl, but nothing can be done without evidence. The evidence you need can be found on the vampire itself who is usually sitting around the court just outside where the Jarl is. Go and pickpocket the vampire if your skills are not that high. You can use potions such as a Potion of Invisibility to help you, although it should let you pickpocket the visiting advisor even without consequences even if you are marked as "detected".

Once you have pickpocketed the orders, you need to bring the evidence to Skald. Tell him his is in danger, a vampire has infiltrated his court (give letter). The Jarl tells you to take care of this unwanted visitor, and confirms his guards will not interfere with you carrying out justice.

You now need to destroy the vampire masquerading as a Visiting Advisor. You can now safely go and kill him wide in the open. The guards will not interfere, they might even help you.

After you have killed the visiting advisor, return to Isran at Fort Dawnguard. Tell Isran you have destroyed the vampire in the Jarl of Dawnstar's court.

This completes the quest, A Jarl's Justice.