First things first: in order to mine Stalhrim deposits, you'll need a special pickaxe called the Ancient Nordic Pickaxe. The easiest way to get one: When you arrive in Raven Rock at the beginning of the DLC, speak to the blacksmith Glover Mallory and he'll give you a task to fetch one for him. Go to Raven Rock Mine, convince Crescius to give you the pickaxe, and then return it to the blacksmith. He'll let you keep it.
In order to learn the ability to craft using Stalhrim, make your way to the Skaal Village in the northeast part of Solstheim (the main story of Dragonborn will take you there at several points, and you may need to progress a bit in the main story in order to open up the needed sidequest). Talk to some of the residents and you'll learn that the town's blacksmith, Baldor Iron-Shaper, has been abducted. This will begin the A New Source of Stalhrim sidequest.
At the end of this multi-part sidequest, Baldor will teach you to craft with Stalhrim. If you need some of the material itself, it can be found in various locations around Solstheim, but you'll also receive a map at the end of the sidequest that points you to a location with a large amount of the material for mining.
Important note: Stalhrim is tied to Ebony in the game's smithing system, so in addition to the above you must also have the Ebony Smithing perk unlocked in the Smithing perk tree (requires 80 skill level in Smithing) in order to get this achievement. However, the DLC allows you to respec any of your previously-selected perks at the end of the main questline, so don't worry if you have a maxed-out character and haven't unlocked Ebony Smithing.
UPDATE: Some people have reported this achievement as missable. Whether it's a bug or not, here are some things you need to make sure you do so that you don't have to replay a lot of the DLC to get this one:
- When freeing the Skaal from Miraak's control at the Wind Stone, make sure none of them die, especially Fanari Strong-Voice and Deor Woodcutter.
- As soon as you reach the point in the DLC's main storyline where the Skaal are freely moving about their village (after The Fate of the Skaal quest), talk to Fanari Strong-Voice and she will give you a Miscellaneous task to find a bandit leader. Complete this task as soon as possible, before progressing further in the story.
- Return to Skaal Village and you'll soon overhear a conversation outside about the missing blacksmith (use the Back button to wait if necessary). Talk to Deor, and the quest to find Baldor will begin.