It's always been confusing to access the downloadable content in the Souls games, and this tradition continues with Bloodborne. It's not as arcane as it was in the past, but there are steps you should be aware of -- it's not just a menu.
First, you'll need to make your way to the Cathedral Ward, one of the early hub sections in the game. Head up the stairs to Vicar Amelia and kill her (or try!).
From here, you can trigger a cut scene by interacting with the altar, which both opens up a new section in the main game and allows you to find the DLC.
Next, head to the Hunter's Dream, spend some blood echoes, and pick up a item called the Eye of a Blood-Drunk Hunter from a messenger on the floor.
Head back to the Cathedral Ward, then go outside the door to the left.
Look for a graveyard on the right side. Look closely -- you'll notice a guy hanging next to a tomb. Give it a moment and a horrifying creature will pick you up.
In regular Bloodborne, you'd die here and wonder what the hell happened. But now, you're ready for the Hunter's Nightmare. Good luck and happy hunting!