You will receive a piece of the Pirate set after completing Isle of Lost Skulls. There are two differently named sets, the Pirate Captain set (Red/Black/Gold sets), and the Scallywag sets (Red/Brown/Black sets).
The item you receive depends on your quest rating at the end of the quest.
C rank: Cuffs / Belt (Brown/Red/Black/Red)
B rank: Belt / Shirt (Brown/Red/Black/Red)
A rank: Shirt / Hat (Brown/Red/Black/Red)
S rank: Any piece from the Gold set
As with any quest, once you have obtained every piece of the black set, you will receive a piece of the lower ranks you haven't gotten yet.
All cosmetic pieces except S rank items can be bought with gold you earn during the quest, in the lobby Gift Shop.
In a recent update, a new outfit was added to the set, which is the Black Scallywag set. Much like the Aristocrat outfits in Crescendo of the Blood Moon, it can only be purchased through the Swash and Buckle Gift Shop. Each piece is quite expensive, you need 40,000 Gold to get it, This outfit can not be obtained through beating Isle of Lost Skulls, no matter what rank.