STEP 1- Acquire Husk from Killing Hive Enemies
STEP 2- Upgrade the husk by killing hive (see spots below for thrall, knights and wizards)
Step 3 - Upgrade and Speak to the Speaker
Step 4- Use 25 Motes of Light to purge the Darkness and upgrade Eidelon Ally
Step 5- Talk to Eris to receive the quest for Essence of the Oversoul
Step 6- Traverse the Abyss, Kill Crota 390 and Kill Omnigul. To receive the Essence of Oversoul
Step 7- Use the Essence on the Final node on Eidelon Ally to upgrade it to Necrochasm.
Enemies with yellow bars will not fill the HotP progression bar.
Named enemies (even with red bars) will not fill progression bar.
Taken enemies will not count.
You can transfer both the Husk and Eidolon Ally between your characters to progress through the 2 first steps of the quest on all your characters at the same time.
The number of kills needed seems to vary, but would be of around 20 wizards, 70 knights or 350 thralls if you farm in a story mission. Numbers appear to go up if you farm outside a darkness zone.
The Dark Beyond : Fast and easy, this good old method (as old as Destiny itself) consist in running through the mission until you revive a fallen guardian. Once the cutscene over, kill as many thralls as you can (there will be quite a lot of them), then die and repeat.
Siege of the Warmind : Also an old farming spot. Run through the mission until you get in the Bunker. There will be a lot of thralls in here, kill them all then jump of the map to kill yourself and repeat the process. This is probably even faster than the previous method.
The Wakening : First of all, kill all knights on your way (9). Then after meeting with an ogre, there will be 4 knights kneeling in front of the Crystal of Crota's Soul. Kill them, die and repeat killing these 4 knights again and again.
The First Firewall : If you (still) have access to this mission.
Dreadnaught Patrol : Run around the Court of Oryx, there's a bunch of knights spawning here and there.
Earth Patrol : Go to Rocketyard, 4 knights are in this area and they respawn fairly fast.
Moon Patrol : Near the beginning of the Summoning Pits Strike, right at the entrance of the undergrounds, 3 knights spawns quite often there.
The World's Grave : If you're an adept of the "Rinse'n'Repeat" Method, you can run through that mission until the very last room, kill all knights (there should be 1 outside the room and 2 inside), die and repeat (you will respawn not far from that room).
If you play with a friend who needs thralls, you can stick together and do this missions since there's a bunch of thralls at the same checkpoint.
Chamber of Night : If you're more into running though a mission crushing everything on your road, this a pretty good mission : I counted a good dozen of knights. Only downside : it actually take some time to encounter your first knight.
Siege of the Warmind : Same checkpoint as for thralls, just go for the knights (2 of them spawn here).
The World's Grave : Run through the mission until you have to kill a knight to get a key. A wizard will be with him in the room. Once killed, turn around, the hive ships outside will have drop another wizard (sometimes 2). Jump of the map to repeat the process. As of now, this looks like the fastest option.
Lost to Light : Beginning of the mission, you'll get in a darkness zone. There's one wizard, kill it, die, repeat. Simple.
Dreadnaught Patrol : When running around Court of Oryx, you have a chance to meet with wizards. From what I've heard, when standing in the CoO room (where you spawn at the Beginning of Oryx Raid), go on one side (like if you'd go grab the orb) and there should be a wizard, kill it, then run to the other side and if everything's going as planned, a wizard should be waiting for your wrath. Once dead, the first one should've respawned, so just repeat the cycle.
Chamber of Night : Run through the mission until you get to a room where a ship will spawn dropping a wizard and a bunch of acolytes. Kill the wizard and... you get it, "Rinse'n'Repeat" !
The Last Array (LL240) : First of all, put the mission at Light Level 240. Run all the way to the end of the mission where hive ships starts appearing. 2 wizards will spawn from the first ship, kill them and jump off the cliff to repeat the process.
Moon Patrol : Go underground and make your way from one side to another killing every wizard on your way. Might be a bit hard to orientate yourself if you're not used getting down there, but you'll eventually make it.
Summoning Pit (Level 10) : On the easy version of the strike , at the Final Boss arena, just stay to that little room you just came from and destroy the shrieker. Eventually a group of 3 wizards should come for you.