To create a Fire Weapon you need to begin with a standard weapon ( one that you can appply pine resin to otherwise it is unique and will not work )
Then you will need to recover the Large Flame Ember which is located in Lost Izalith and give this to Vamos who is located in the catacombs which will allow him to complete his Fire upgrade from +5 to +10. Vamos can automatically make Fire weapons to +5.See vid for location video by Marce1024
In order to complete the Fire weapon from scratch you will ultimately require 9 Titanite shards , 11 Large Green Titanite shards , 9 Chunks of Red Titanite and 1 Red Titanite Slab
Red Titanite Chunks - Lost Izalith the tentacle monster plants can drop them and they respawn apparently but the best place to farm all chunks are at the very end of the game in the Kiln of the First Flame.
Red Titanite Slab - Rare drop from Chaos Eater in Lost Izalith
and in a treasure chest prior to boss Bed of Chaos(the Giant Tree) of Lost Izalith(see vid credit to MrSiegeBreaker13).
Here is a Chart which was created by an unknown so credit to whoever they are.
TRICK-To get the achievement without using your items simply fully upgrade the item in one shot and wait until the achievement pops WITHOUT leaving the upgrade window thus not allowing the game to save the data then simply turn off the game and turn back on. Supposedly worked on Demons Souls
***thanks to NITEMARECREATOR for correcting and verify the number of Large Green Titanite and Chunks to be used.