Kill 50 Majors or 10 Ultras (Ultras like Zydron at Eye of the Gate Lord on Venus count for 5 a piece) – I went to the Exclusion Zone mission on Mars in hard mode which spawns 9 yellow bar Majors, do this mission over and over till you get all the kills done.
Go back to Shaxx and he will let you choose your Arc, Solar, or Void Legendary sword.Collect 25 Hadium Flakes from opening chests on Dreadnaught Patrol, Motes of Light, and Planetary Materials.Eris will give you A Broken Will quest, and have you talk to Lord Shaxx and he will give you a Reforging the Blade quest.Complete The Taken King story missions to Unlock the Dreadnaught Patrol Area.Keep in mind that whatever Legendary sword you choose from Lord Shaxx is the element damage Exotic sword you will end up with, also you get to keep both the Legendary sword and the Exotic sword when you complete the quest line.
This is an in depth guide on the quests and steps that need to be completed in order to get a Exotic sword like the Bolt Caster, Raze Lighter, and Dark Drinker.