To upgrade the Leviathan Axe to it’s maximum level 6 and get the “Worthy” trophy, you need to collect the Chilling Mists of Nilfheim and trade it for a Frozen Flame. All previous upgrades to level 5 can be done by following the main story and collecting the frozen flames from boss fights.
Travel to Nilfheim and talk to Sindri who has his shop just before the maze. Collect his quest “Ivaldi’s Workshop” to open the center chamber and collect 500 mist echoes. These can be found inside the mist in chests.
When you enter the mist you can see a red bar on top that is slowly decreasing. When it is empty you die and lose your progress of your current attempt. The bar can be filled up slightly by opening chests.
To save your progress go back to Sindri’s shop. After colelcting 500 mist echoes, buy the entry stone and open the door to the center chamber.
To upgrade your leviathan axe you need the Chilling Mists of Nilfheim that is inside the second chest from the left. It costs 5000 mist echoes to open so you will need to run a couple more rounds in the maze.
Once you have enough mist echoes, collect the Chilling Mists of Nilfheim and use it to buy a frozen flame from the shop. Now you can fully upgrade your leviathan axe to level 6 and get the Worthy trophy.