In order to begin the first quest in the Dragonborn DLC that will lead you to the island of Solstheim, you must have progressed in Skyrim's main storyline to at least your first visit with the Greybeards, completing The Way of the Voice quest. After meeting with them for the first time and receiving The Horn of Jurgen Windcaller quest, you're ready to officially begin Dragonborn. If you're using an existing character that has finished the game's main quests, you're already set to go.
Make your way to nearly any of the game's main cities (Whiterun works perfectly well) and you will be almost immediately approached by some Cultists. Listen to what they have to say, defeat them, and read the Cultists' Orders note on one of the bodies. This will begin the Dragonborn quest. You'll be directed to the docks outside Windhelm, where a boat awaits you. Talk to the captain and either persuade, intimidate, or pay him 500 gold to take you to Solstheim. After a short cutscene, you'll arrive at Raven Rock and the achievement will unlock.
Note that this achievement can actually be obtained right after leaving Helgen at the beginning of the game by visiting the docks at Windhelm and paying the captain 250 gold to take you to Solstheim. However, you will not be able to progress through the main questline of the DLC until getting the Dragonborn quest from the cultists, so it's better to use an existing character or create a new one and progress to The Horn of Jurgen Windcaller if you want to tackle the DLC in its entirety.