In order to make passage to the island, you need to 1) be best pals with your friendly neighborhood Synthetic-Man, Nick “The Detective” Valentine and 2) you need to have completed his “Get A Clue” quest.
After that, another quest called “Far From Home” will pop up and a new radio signal will direct you back to Valentine’s Diamond City detective agency to pick up a new case.
From there, you’ll be sent to the far northeastern corner of the map to meet with the Nakanos, a couple who desperately want you to locate their missing daughter. Once you agree to help them, hop on that big fishing trawler, and you’re on your way to the fog-filled freakfest that is Fallout 4: Far Harbor.
And while you’re at it, check out Bethesda’s vidoc about the DLC below. Did you know it’s a bigger island than any previous DLC they’ve made for any other game? That’s huge.