Talk to Gunmar in Fort Dawnguard and ask him what you can do to help. Gunmar tells you he has learned of a possible monster, and while he does not have information on the vampire's exact whereabouts, he has information suggesting someone who does. Pay a visit to the beast's last known contact. Do what you must to find out where it is hiding now. Then seek out the monster, and destroy it.

This begins the quest, Hunting the Monster.

Open your quest log:

Hunting the Monster
Gunmar has sent me to Winterhold where I am to learn the current whereabouts of the Master Vampire. I will learn this through interrogation of his last known contact Birna, or searching for evidence. Once its location is obtained, I am to destroy the beast in its lair

Find evidence of the Vampire's whereabouts.

Note: The vampire's last contact is assigned at random. Other than Birna, possible contacts include:
Nenya (Falkreath)
Hamal (Markarth)
Susanna the Wicked (Windhelm)
Carlotta Valentia (Whiterun)

Travel to Winterhold and find Birna's house or shop (Birna's Oddments). 

Note: Sometimes the marker will appear bugged and point to the second floor instead of the first. Go down to the first floor and locate the NPC.

Talk to Birna and tell her you are looking for a vampire. She does not know what you are talking about. Your options are:

Make this easier for you, tell me what I want to know (PERSUADE)
Perhaps some coin will jog your memory (400 gold) (BRIBE)
I'll get rough with you if I have to (INTIMIDATE)

If your speech skill is low, then use BRIBE since that never fails. Otherwise, use a PERSUADE or INTIMIDATE to get the information you need.

If none of these work, you can try pickpocketing her for the evidence.

She gives you a Letter from the Vampire. That letter is the last she has heard of him and mentions where he is.

Quest description updates to:
Gunmar has sent me to Winterhold where I have learned from Birna the whereabouts of the Master Vampire. I am now to journey to Stony Creek Cave and destroy the beast.

Your objective updates to:
Kill the Master Vampire at Stony Creek Cave

The location of the Master Vampire is also random. Other than Stony Creek Cave, you might get sent to:
Halted Stream Camp
Hob's Fall Cave

Go to the dungeon that is mentioned in your quest log, and find the Master Vampire. Enter the dungeon, find the Master Vampire, kill it, then return to Gunmar in Fort Dawnguard. Tell him you have destroyed the Master Vampire.

This completes the quest, Hunting the Monster