This mission becomes available only after you reach 36th level. In order to begin it, you should read a book called Deathbrand - you can find a few copies of it in Solstheim. Their locations are as follows:
Steward's House in Tel Mithryn - on a shelf by the bed;
Fort Frostmoth - on a shelf in the room where General Falx Carius is;
Haknir's Shoal - in the equipment of one of the pirates;
Morvayn Manor in Raven Rock- in the upper floor bedroom, by the bed;
Temple in Raven Rock - on the lower floor, in a room on the right, on a table;
Tel Mithryn - on a shelf by the magical "elevator" and on one of the tables;
The Retching Netch - in the bedroom in the northern part of the building, on a shelf;
Skaal Village - in the Greathall.
Pick up the book to activate the mission - but only if you have reached level 36. Alternatively, you can head to Haknir's Shoal and find the Deathbrand Helm in a chest - that way you will avoid the part with searching for the books.
Use the Deathbrand Treasure Map to find the treasure
Using the treasure map which you found by the pirate leader in Deathbrand Helm, you shouldn't have problems with finding four parts of the ancient armor. Below are the locations:
Boots - in a chest west of Tel Mithryn, on the shore;
Helm - in a pirate camp in Haknir's Shoal, in an ancient chest;
Gauntlets - in a ditch south of Northshore Landing (see: A New Source of Stalhrim), opposite to the Rieklings camp, beside Bristleback Cave;
Armor - in a chest north of the Earth Stone (on the shore).
After obtaining the last part of the set, in the chest you should also find a key to Gyldenhul Barrow. That's where the last part of the mission will take place.
Explore Gyldenhul Barrow
Gyldenhul Barrow can be found east of Skaal Village, on an island. Upon reaching this location, you will be attacked by Bandits - they are however the least challenge that you will face. Head inside the building, dig through the lore of Stalhrim found on the right (if you don't have a special pickaxe, complete Retrieve the Ancient Nordic Pickaxe).
Head through the mysterious corridor hidden behind the wall of Stalhrim, go through the gate and enter the dungeon. In the room in which you will arrive you can find lots of treasures - gold and other useful items. If you try to pick up any of them, the gate behind you will immediately shut - apparently there's no way back. The only thing you can do now is head inside the tomb of Haknir Death-Brand.
Defeat Haknir Death-Brand
Behind the throne of the dead pirate you will find an unique sword - the Bloodscythe. Pick it up and the ghost of the pirate will appear in the room and immediately attack you. Defeat him as fast as possible, as otherwise he might call in a few minions - members of his crew. Luckily you don't have to fight them - only Haknir is the required target (his minions will disappear when you kill the Captain).
After killing the enemy, take the second sword - Soulrender - from his corpse. It's a very useful weapon which might come in handy.