To create a Strongest 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 Locate the Very Large Ember in New Londo Ruins AFTER you lower the water in the region (See vid below for location and credit to Marce1024 for the vid) and give it to Andre of Astoria this will allow him to upgrade your weapons from +11 to +15.
In order complete the Strongest weapon you will ultimately require 9 Titanite shards , 9 Large Titanite shards , 8 Chunks of Titanite and 1 Titanite Slab
Titanite Chunk - 100% drop rate off the first and third Black Knight in Kiln of the First Flame. These enemies do respawn
Titanite Slab - is a VERY rare drop off Darkwraiths in
New Londo Ruins and also a 100% drop off of the Stray demon in the Undead Asylum (tutorial area) underneath the place you fought the Asylum Demon (first boss about 2 min into the game). The asylum can be revisited via the massive crow who you can see at Firelink Shrine who brought there in the beginning of the game.
Here is a Chart which I believe someone named Oyabunn created so credit to him.
If you are looking to gather materials for every weapon upgrade achievement you will need a combined total of 90 Titanite Shards , 20 Large Titanite Shards , 24 Chunks of Titanite , 3 Titanite Slabs , 60 Large Green Titanite Shards , 16 Chunks of Blue Titanite , 2 Blue Titanite Slabs , 16 Chunks of Red Titanite , 2 Red Titanite Slabs , 16 Chunks of White Titanite and 2 White Titanite Slabs
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.