This is NOT my solution - this is a copy/paste from BiggDope's solution back from 02 June 2016 for the non GOTY edition of the game.
Link to original solution:
(contains a video as well)
This achievement is obtained by completing the side quest entitled “There Can Be Only One” and is highly missable.
The quest can be triggered by reading the “Test Yourself with the Trials of the Virtues” note on any of the notice boards within the city of Beauclair.
Once the quest is active, head over to the isle on Lac Celavy and speak with the Hermit. You will learn that in order to gain access to Aerondight you must first display the five chivalric virtues during your stay in Toussaint: Compassion, Generosity, Honor, Valor, and Wisdom.
Depending on when you choose to tackle this quest, you may have already completed some of the tasks and/or quests that would have already displayed such virtue. There are about three instances for each virtue, all of which are usually masked by timed dialogue choices you must make during quests. Make sure to read the inscriptions on the stones near the Hermit in order to get some clues for the virtues you might be missing.
Please see below for a checklist of earliest tasks and/or quests you will come across in Toussaint. For additional opportunities throughout the expansion, visit THIS LINK.
Trial of Compassion
Main Quest: The Beast of Toussaint - After fighting the Shalemaar in the arena, you must decide to spare its life.
Trial of Generosity
Miscellaneous Task: Beauclair - After completing "Blood Run," enter the Main Square of Beauclair and a child will deliver a letter to you. Give him 5 crowns for his trouble.
Trial of Honor
Side Quest: The Warble of a Smitten Knight - After the horse race, choose to accept the mounted duel against the Rivian Prince and defeat him.
Trial of Valor
Side Quest: The Warble of a Smitten Knight - After following the Oriel (yellow bird) into the woods and returning to camp, you must decide to finish the tourney and go on to win.
Trial of Wisdom
Main Quest: La Cage Au Fou - Thoroughly explore the Wight's Lair and when hiding in the closet, try to lift the curse instead of attacking the creature.
As mentioned, there are a few more instances to display each virtue throughout the expansion, but the ones listed above are those in which you will come across the earliest. If you happen to miss any, you can choose to carry on and nab the other opportunities, or start the expansion from scratch from the main menu with a pre-made Geralt. It should only take about half an hour to run through the aforementioned quests and obtain all the virtues.
Once all the trials have been completed, return to the Hermit on Lac Celavy. To reclaim the coveted sword, you must first defeat the Hermit. Afterwards, the Lady of the Lake will appear and award you Aerondight, thus unlocking the achievement.