This is a somewhat lengthy, multi-step process, and only possible in the post-credits chapter of the game. The steps do not have to all be done in this precise order, but I'm arranging it this way to try to make it as easy as possible.
We're going to be forging the Super Sword of Light.
Step 1: Obtain the Forging Hammer - It is in Gallopolis, as a prize for winning the final tier of Easy horse races. The racetrack can be found by running down the loooooong tunnel in the middle of the city.
Step 2: Obtain Orichalcum from Battleground - Just like forging the original Sword of Light, go to Battleground, and grab the ore from the same spot it was before for story progression.
Step 3: Go to Trial Isle - If you have not found this place yet, continue advancing the endgame story, and you will. While here, do a lap around the perimeter of the island and grab Orichalcum from the chest here. (counts as different from the story one.)
Now, enter the first trial. Over the course of getting to the end of the trial, you will enter the Eyrie. While here, go to an area marked as the Phoenix Nest on the map. The chest here contains Lumen Essence. We'll need that later.
At the end of the trial, you will get to make a wish for clearing the final boss fight. Wish for a powerful weapon. (You can get the other wishes later). This will give you the recipe for the Super Sword of Light.
Step 4: Obtain the Sword of Kings - Travel to Hotto. Head left up the stairs, and there will be a man outside an item shop who wants to buy that treasure chest Orichalcum off you so he can make the Sword of Kings. Agree to sell it. Now leave Hotto, and re-enter. Talk to him again, and he will tell you the sword is forged. Go inside the item shop behind him, and buy it for 35,000 gold.
Step 5: Obtain the Crucible Key - Go to the Volcano area near Hotto, and enter the dungeon where you had to fight Ryu (the kid trapped in dragon form). Get to that boss room, and his mother is there. She will ask for the Lumen Essence to power the mirror. Give it to her to restore her son. Back in Hotto, talk to her to obtain the Crucible key as thanks.
Step 6: Forge the Sword of Light - Same process as it was for the story. The crucible is the locked door in the Hotto Volcano. Go there to trigger the cutscene.
---The Super Sword of Light recipe calls for Crimsonite, which cannot be purchased. Time to get some! ---
Step 7: Obtain Serpent Souls - Go to the Ruins of Dundrasil, and fight the Royal Reptiles in there. (Biped, multiple tails) You can either steal the Serpent Soul from them, or they will drop it as an item. I'm sure other mobs drop these, but I stole them very easy here. You need two of them.
Step 8: Go to Sniffelheim Castle, and talk to Krystalinda in the throne room. Exchange those two Serpent Souls for Crimsonite.
Step 9 - Forge the Super Sword of Light! AT LAST! - Quality does not matter for the achievement. You can get "Failure" status, and the sword will still be made, and the achievement will still unlock.
Sorry about any misspellings on items / locations - guide should be clear enough that you'll know what I mean. If leaving a thumb down, please let me know how this could be improved.