When you reenter Drangleic after downloading Crown of the Sunken King, you'll find a new item in your inventory. Though Dark Souls 2 won't say so, the Dragon Talon is your key to this new area. There are a couple of ways to get there, depending on where you are.
IF YOU CAN WARP TO BLACK GULCH
Head to Hidden Chamber, the second bonfire in Black Gulch, and walk to the room where you fought The Rotten.
IF YOU CAN'T WARP TO BLACK GULCH
The quickest path to Black Gulch is through The Gutter, which you can access from the well in Majula. Equip an item that minimizes fall damage like the Silvercat Ring or use the Fall Control spell and hop from platform to platform. Make your way to The Gutter's ground floor and enter the mist that takes you into Black Gulch. Make your way through the toxic landscape and defeat The Rotten.
FROM BLACK GULCH
In the back of The Rotten's former lair, behind a small fire lake, there's a doorway. Take the path to the room with the Primal Bonfire. (While you're at it, make sure to open the treasure chest containing an Estus Flask Shard, which lies in a small branching path on the right side of the tunnel just beyond its entrance.) Behind the bonfire, you'll see a monument wreathed in snakes. Examine it, and you'll warp to Crown of the Sunken King's introductory area, Shulva, Sanctum City.
Walk past the fire, open the door and down the hall. In the hallway, to the left of the rock formation just past the room with luminescent crystals, turn left to find a corpse (pictured below) with Dragon Charm x 3.