This questline is huge. You have to collect 45 out of 50 calcified fragments, beat the entire King's Fall raid and play many missions. However, at the end of the day you will earn one of the best weapons available at the moment; the Touch of Malice, an exotic scout rifle which is really amazing.
Ok, so here are the mainsteps and some little explanations. It may happen that these steps come at you in different order depending on what you have already done (collect / visit) in the game.
You can check how many fragments you have collected while being in Orbit when you select the destination Saturn -Dreadnaught.Step 1:
Collect one Calcified Fragment and return to Eris Morn in the Tower
You will find Calcified Fragments only in Dreadnaught areas, there are 50 fragments total in the game. For a location guide of 46 / 50 fragments (4 fragments still undiscovered as of 09/27/2015) have a look at the end of this solution.Step 2:
Eris Morn will give you the first quest called Shattered Past which requires you to collect 5 more calcified fragments. After you have found these 5 fragments, return to Eris Morn.Step 3:
The next quest called Hunger Pangs with the Step "Promethean Code (heroic)" is unlocking after you have collected 15 calcified fragments.
Go to Bunker War-4 and kill the Taken Centurion at the end of the mission, after you have done so, go back to Eris Morn.Step 4:
Collect 30 calcified fragments and the Hunger Pangs The Undying Mind quest is unlocking.
Play in this special version of the Undying Mind Strike until you reach the final boss fight against the Undying Mind. Shoot the boss' health down to 50% to trigger the second reinforcement wave. Morgath appears with these reinforcements. Morgath should appear near one of the little caves at the right side of the boss area. The problem is, the Vex fight against him so you have to be fast enough and kill him before the Vex kill him (If you fail, restart the boss fight). Step 5:
After you have the Essence, talk with Eris in the Tower and go inside the King's Fall raid to trigger the next quest called The Old Hunger Form the Ritual. (Go in Raid and leave directly to the Orbit, thats enough)
For this quest you have to defeat the whole King's Fall raid Bosses (Warpriest / Deathsingers and Oryx)Step 6:
After you get all 3 items, talk with Eris again to get the next quest step. Here she needs you to collect 25 Wormspore, 50 Hadium Flakes and 50 weapon parts for her.
Wormspore and Hadium Flakes (only in random chests) can be found in all Dreadnaught areas in the game, the best way to do this is patrol mode and farming chests and wormspores. You obtain weapon parts when you dismantle weapons.Step 7:
Collect 45 fragments to unlock the next quest step called Hunger Pangs Fear's Embrace.
The last mission in on Phobos, besides Mars. Play the mission until you reach the circle room with the Echo of Oryx inside, shoot down the Echo to 25% and the wizard appears. Kill the wizard and get the essence. Return to Eris for a last time and claim your Touch of Malice. Congratulations, Guardians.
Here are the video guides which help you to solve this long quest:
first, the complete guide (the steps are not the same as here in this solution)
second, the Morgath step shown in a detailed way
lastly, the calcified fragments (in the spoiler is the timeline with timelinks)