In order to play Dragonborn, you will have to meet a few conditions. First of all, you of course have to download the expansion and launch the game on the account assigned to it.
The new quest will become available only if you have already completed The Way of the Voice main mission, meaning you've reached The Horn Of Jurgen Windcaller.
If you've already done the above, head to any city in the game and wait for two cultists (that way you will begin the main story of the expansion). Alternatively, you could head to Windhelm and speak with a man named Gjolund Salt-Sage at the docks. He will take you to a new land, though you won't activate the story itself (of course there's always a chance you will meet the mentioned cultists there and everything will go well).
Find out who sent the cultists
The Cultists will ask you if you really are the Dragonborn. Regardless of what answer you give them, you will be attacked. The fight itself isn't really hard - the local guards will probably come to help you. The enemies aren't too tough, though they can cast some powerful spells - keep that in mind. After dealing with the threat, search their bodies to find the Cultist's Orders. Pick it up. If you're running low on cash, you can always sell the rest of their things - they're worth some gold.