This achievement is fairly straight forward. Upgrading your weapons involves using various items of Adamantine to level up your weapon. The achievement popped for me when I reached level 10 on my weapon but I don't know if the Crucible DLC impacts this as it is possible to level up and 11th time with it. Below I've made a chart of the 11 levels and what items they require.
Level 1: 2 Sliver of Adamantine
Level 2: 4 Sliver of Adamantine
Level 3: 6 Sliver of Adamantine
Level 4: 2 Fragment of Adamantine
Level 5: 4 Fragment of Adamantine
Level 6: 6 Fragment of Adamantine
Level 7: 2 Piece of Adamantine
Level 8: 4 Piece of Adamantine
Level 9: 6 Piece of Adamantine
Level 10: 1 Chunk of Adamantine
Level 11: 1 Wicked Adamantine
In total you need 12 Slivers, Fragments, and Pieces of Adamantine, 1 Chunk of Adamantine, and possibly 1 Wicked Adamantine if you have the Crucible DLC.