This is just what I did.
***Edit*** To begin the dragonborn quest you must first at least start though some say complete the main storyline quest 'the horn of jurgen windcaller' and then wait in a main town for the cultist attack.
Thanks to Subliminal Cuts for the info.
When you arrive in Solstheim and have to find out about Miraak turn around and go across the water to where there is some people in a trance building and a man watching.
Talk to this man and he will direct you to the temple of Miraak.
Go here and talk to frea and she will tell you to find a way into the temple.
Once the steps open up go down into the temple.
From here its a simple case of moving through the temple killing mostly Dragur Deathlords and Dragur Wights and the ocassional cultist.
In the room with the first dragon aspect kill the dragur and search the bodies for a key.
There is a door in one of the coffins then just continue on until
you get to the last room with the first black book.
Read it, Cutscene, Achievement Unlocked.