To unlock the Exotic quest, level up your legendary sword completely - purchasing all of the upgrade nodes on the weapon's upgrade screen. When that's done, you need to infuse your sword beyond Light level 280. To do this, save up your higher level Rare (blue) Heavy weapons and infuse them into your sword. Once you meet those requirements, talk to Lord Shaxx (the Crucible Handler) in the Tower, and he'll present you with the Exotic quest.
Your first objective is to kill 50 major or ultra enemies (with a yellow health bar) and 25 Guardian in the Crucible using your sword. Run Heroic Strikes and jump into objective-based Crucible matches, then start slicing and dicing. Once you've finished your PvE and PvP killing sprees, return to Shaxx.
Next, you need to kill Ecthar, Sword of Oryx. To find him, drop into a Patrol mission in the Dreadnaught and head over to the Founts section of the map. You'll need to drop into the lower level and travel to the central hall, where you will find three Warden's Knights. In order to progress, you need to kill all three Knights within a few seconds of each other. The easiest way to do this is to whittle down their health, then finish them off with a Super ability (like the Titan's Hammer of Sol or the Warlock's Stormtrance). Kill the Knights in quick succession and continue through the now-unlocked door.
Next you'll face Ecthar, a Hive Knight with a white shield. The only way to damage his shield is to break it with your sword. Hack away, and don't forget about blocking his sword strikes. When you come out of your epic sword duel victorious, head back to the Tower and talk to Shaxx.
The Crucible Master will send you on a quest to collect materials from resource nodes, based on the sword element you chose; You'll also need to score some Arc, Solar, or Void ability kills. The Solar sword requires Helium Filaments from the Moon, the Void sword takes Relic Iron, and the Arc sword needs Spinmetal from the Cosmodrome. You'll need to farm 10 special materials (called Zeptocyte Cores for Arc, Solitron Flares for Solar, or Relic Crystals for Void) from the resource nodes (not the chests) on your respective planet. To farm the ability kills, replay the first section of the Siege of the Warmind mission. The hall before the large Omnigul room throws dozens of thrall at you. Kill them, die on purpose, rinse and repeat.
When that arduous task is over, talk to Lord Shaxx on Armsday better known as Wednesday. Shaxx will then give you a quest that requires you to complete a special version of the Sunless Cell strike. In it, you'll need to slay a specific Warden and the Strike's boss Alak Hul within 30 seconds of each other. Get them both down to low health, then unleash hell on your Warden, quickly transitioning to Alak Hul after the Warden falls. It has to be done in this order, or it won't work.
Now you're done! Head over to Shaxx and claim your new Exotic sword.
