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Perform all forms of infusion.

To infuse a weapon you need to take it to Blacksmith Andre in Firelink Shrine, you most have to relevant coal and the necessary gem stone. To get them all you'll need the following:

Farron Coal - found in the Road of Sacrifices. From the Halfway Fortress bonfire go straight ahead and drop down the ledges on the right side, past some poison beasts and crucified undead and into a door at the end. Hand a right here and head down to a Black Knight the coal is on a body just past him. Allows Heavy, Sharp and Poison infusion.
Sage's Coal - found in Farron Keep. From the first bonfire head left through the swamp until you come to a tower guarded by a Darkwraith, the coal is on a body inside. Allows Crystal, Blessed and Deep infusion.
Profaned Coal - found in Irithyll Dungeon. In the same large room where you find multiple jailers and Karla the Witch is locked up. On the left side (facing the main exit) is a room full of malformed beasts, the coal is on a body in this room. Allows Dark, Blood and Hollow infusion.
Giant's Coal - found in Anor Londo. From the bonfire, head up the stairs and hand a left. Enter the doorway through here to find the coal on the giant blacksmiths corpse. Allows Lightning, Simple and Chaos infusion.

You'll also need the relevant gems which can be found in a few places or dropped by enemies, but here is a helpful guide:

Fire Gem - can be chosen as a starting gift or dropped by the Fire Demon you fight with Siegwald in the Undead Settlement.
Refined Gem - on a corpse between the two dragons in Lothric Castle or dropped by a Crystal Lizard in the Grand Archives.
Heavy Gem - speak to Hawkwood at Firelink after you kill the Crystal Sage or dropped by Crystal Lizard in ravine (on way to find Irina) in Undead Settlement.
Raw Gem - kill Crystal Lizard on roof with Pus of Man enemy in High Wall of Lothric or can drop from the hollow enemies in this area.
Crystal Gem - kill Crystal Lizard in the building before Crystal Sage boss fight or dropped by Sorceror Hollows in Crucifixion Woods.
Hollow Gem - from the Farron Keep Perimeter bonfire turn around and head forward until you reach a hole in the wall to your right, drop down to a ledge with this item on a corpse or drop down off the roof in the Grand Archives (after the first Gargoyle) and find it on a corpse in a corridor behind you.
Deep Gem - in Cathedral of the Deep, head around the room with the giant and up the stairs, then go left up the next stairs and head up to a balcony with an Evangelist enemy the gem is on a corpse here. Also dropped by Deacons of the Deep on your way to Anor Londo or in Anor Londo itself.
Blessed Gem - in Cathedral of the Deep make your way up to the rafters above the giant and head to the right side, it's behind a Knight with a Greatsword, also in the rafters of Lothric Castle if you drop down the hole in the building after the three Gold Winged Knights.
Poison Gem - from the Keep Ruins bonfire in Farron Keep go forward and look right to see three large Ghru standing around a corpse holding this item or grab from a corpse in the toxic swamp area in the Profaned Capital.
Sharp Gem - kill a Crystal Lizard just above the Dilapidated Bridge bonfire in Undead Settlement or on a corpse by a whip wielding skeleton in the Catacombs of Carthus.
Chaos Gem - on a corpse in Smouldering Lake past all the Giant Crabs (turn off the ballista first) or from a Crystal Lizard near the start of the Demon Ruins in this area.
Lightning Gem - by the large alter in Irythrill of Boreal Valley, just below the Pontiff boss fight door, or one will be on a corpse after the first enemy you encounter in Archdragon Peak.
Bleed Gem - down the steps from the Church of Yorshka bonfire leads to a dark room full of invisible enemies, go through here and drop to a ledge on the right side when you head outside. It's also a rare drop from the invisible enemies.
Simple Gem - in Irythrill Dungeon cross the bridge in the first area and go left, head down some steps to find one on a corpse. Alternatively after the Pontiff boss fight head outside and left around the building, under the bridge and up the stairs. Go back on yourself and head back inside where you fought the boss to find a Crystal Lizard to kill to get a gem.
Dark Gem - in the Consumed Kings Garden look under the left set of stairs just before the boss room or they can drop from Pus of Man enemies here and elsewhere.