You must be at least level 10 to start the Dawnguard quests. Once you have installed the Dawnguard add-on, talk to any guard in one of the major cities (for example a Whiterun Guard) to begin the Dawnguard questline.

Whiterun Guard: Heard they're reforming the Dawnguard. Vampire hunters or something, in the old fort near Riften. Might consider joining up myself.

This begins the quest Dawnguard. Open your quest log:

I've learned of a group of vampire hunters known as the Dawnguard. It seems they are looking for new recruits.

Speak with the leader of the Dawnguard.

Another way to start this quest is to talk to Durak who will come to you, sometimes after you are attacked by some vampires.

Durak: You there, the Dawnguard is looking for anyone willing to fight against the drawing vampires' menace. What do you say?

This gives you the objective: Find out about the Dawnguard.

Your options are:
What's the Dawnguard?
I haven't noticed any vampire menace
Killing vampires? Where do I sign up?
Sorry, I'm not interested.

If you ask him about the Dawnguard, he tells you they are vampire hunters. If you tell him you have not noticed any vampire menace, he suggests you have not been paying enough attention. He tells you about the Vigilants who got tore to pieces by Vampires. He tells you about the fort.

Skyrim Fort Dawnguard Map

Your destination is southeast of Riften. Fast Travel to Riften, then travel south east on the road and past a bridge. It is a pretty long walk there, go around the north side of the mountains. If you have a horse it is a good idea to use it for this trip. Go through a cave to Dayspring Canyon.

You run into a torch bearing NPC named Agmaer who asks you if you are here to join the Dawnguard too. Follow him. Listen to the dialogue, Agmaer also wants to join the Dawnguard and become a vampire hunter. He tells you that you have probably killed lots of vampires and that Isran will sign you right up.

Skyrim Fort Dawnguard

Soon you arrive at Fort Dawnguard which is a very big fort. You might run into Durak again on the way. If you talk to him, you can get him to give you a crossbow. Keep following Agmaer and enter Fort Dawnguard through the main gate. A guard named Celann standing on the stairs in front of the main gate tells you that Isran will decide if you have got what it takes. Enter through big door.

Inside, Isran is talking about vigilants with another NPC called Tolan. Carcette is dead. The Hall of vigilands, everyone, they are all dead. Approach Istan and speak with him.

Your dialog choices are:
I'm here to join the Dawnguard
I heard you were looking for vampire hunters
I was just looking around. What is this place?

Choose the first option. Isran tells you to look around. There is really not much to join yet. He has only started rebuilding the order. Ask him what you could do to help. He tells you that he needs someone in the field, taking the fight to the damn vampires, while we are getting the fort back into shape. Talon was telling Isran about some cave that the vigilants were poking around in. Seemed to think it was related to these recent vampire attacks.

Tolan tells you about Dimhollow Crypt. Brother Adalvald was sure it held some long-last vampire artifact of some kind.

This completes the quest "Dawnguard", and starts the next quest, "Awakening".