This achievement requires having Blacksmith Andre in the Undead Parish upgrade a standard weapon to +15, which would generally require 9 Titanite shards (going from base weapon to +5), 9 Large Titanite shards (going from +6 to +10), 7 Titanite Chunks (going from +11 to +14), and 1 Titanite slab (going from +14 to +15). You'd need to find the Large Ember in the Depths and the Very Large Ember in New Londo Ruins so that Andre could ascend a weapon from +5 to +6 and then from +10 to +11.
Titanite Shards and Large Titanite Shards are found in limited numbers as you progress and can be bought from some NPCs. Titanite Chunks can be found in somewhat limited static locations throughout the mid-to-late stages of the game (as guaranteed enemy drops or on corpses). The first and third Black Knights in the Kiln of the First Flame (the final area of the game) respawn and always drop a Titanite Chunk. You can get two guaranteed Titanite Slabs per playthrough (one from the Stray Demon in the Undead Asylum and one in a chest in Kalameet's area) and a possible third as a reward for completing Siegmeyer's questline.
Standard, Crystal, and Lightning upgrades use the same reinforcement materials. The game only saves when you back out of an upgrade menu, so you can get these three achievements with a single Titanite Slab and 7 Titanite Chunks if you'd prefer to not have to gather 3 Slabs and 21 Chunks. You'll end up losing the resulting fully upgraded weapon by dashboarding right after the achievement pops (before backing out of the upgrade menu) but you will get your upgrade materials back.
You'll want to find two specific weapons: the Lightning Spear in Sen's Fortress and the Crystal Halberd in Anor Londo. These weapons already have the Lightning/Crystal infusions applied, meaning that you won't have to bother with upgrades to get to that point (which would normally require 9 Titanite Shards, 9 Large Titanite Shards, and 1 Titanite Chunk each). You will need the Very Large Ember found in the New Londo Ruins, but you won't need the Large Ember from the Depths since this process does not require ascending a weapon from +5 to +6.
Once you have everything you need, head to Andre in the Undead Parish and give him the Very Large Ember. Start by reinforcing the Crystal Halberd to +5. Wait a few seconds after applying the final upgrade (which requires 1 Titanite Slab) and the Crystal Weapon achievement will pop.
Again, do not back out of this menu screen confirming that the weapon was upgraded to +5, but instead go to the dashboard and force quit the game, then restart the game and continue. You'll be back at Andre as if you had never done any of the Crystal upgrades.
Next, have Andre reinforce the Lightning spear to +5, unlocking the Lightning Weapon Achievement.
Without leaving the final upgrade screen, dashboard, force quit, reload game and continue. You'll be back at Andre as if you had never done any Lightning upgrades.
This time, have Andre "Modify Equipment." The far right option should allow you to revert the Lightning Spear to a +10 Spear. Then choose the option to upgrade ("ascend") the +10 Spear to +11, which requires having the Very Large Ember. Back up to Andre's main menu and choose to Reinforce Weapon, then upgrade the +11 Spear up to +15, at which point this achievement will pop. You can dashboard here if you'd prefer to save your Chunks and Slab for something else.
As always, refer to
http://darksouls.wikidot.com for specific information relating to items or locations.