This requires having Blacksmith Andre in Firelink Shrine upgrade a weapon to +10, which requires Titanite Shards, Large Titanite Shards, Titanite Chunks, and a Titanite Slab. You'll discover these items in increasingly greater numbers as you progress through the game.

You'll need 2/4/6 Shards (12 total), 2/4/6 Large Shards (12 total), 2/4/6 Chunks (12 total), and one Slab to get a weapon to +10.

There are eight Titanite Slabs to be gotten per playthrough:
One from Siegward after freeing him from the prison cell in Ithyrill Dungeon (missable)
One in the Grand Archives after pulling a lever to lower a bookcase
One atop Archdragon Peak on the ground near Havel
One in Archdragon Peak after killing The Nameless King in the area accessible immediately after
One can bought for 30k souls from the Handmaid after defeating all four Lords of Cinder and placing their remains on their thrones
One for killing the three Winged Knights on the roof of the Grand Archives (just before the steps that lead to Prince Lothric boss battle)
One at the bottom of the double elevator closest to the Prince Lothric boss battle
One from trading the Coiled Sword Fragment with the crow at Firelink