To upgrade the Blades of Chaos, you must use a story element known as the Chaos Flame, normally obtained from story fights. The only time you'll need to focus on trying to hunt down a material would be the last one. Muspelheim is where it is located, available from completing the Valkyrie trial and obtaining the "Raging Inferno of Muspelheim" item from the chests. With this item, return to the shop of Brok/Sindri and go to Resources. There will be an option to exchange the inferno for the last Chaos Flame, which you can then use to upgrade the Blades and get the trophy.