Been Waiting a Long Time for This (25 player)

Necrotic Plague was failing to stack in patch 8.1.5 (preventing soloing of this achievement) but it's been confirmed fixed in 8.3.7. If you were having issues during that time, give it a try now! Thanks to everyone who reported/confirmed!

Gonna try to simplify this solo strategy for all classes by combining info.

TL;DR: Stand on stairs, wait for him to summon 9+ Drudge Ghouls, summon yourself a helper mob to attack him so that he'll cast Necrotic Plague on you, let Necrotic Plague jump around until 30 stacks, burn him down.

SHADOWLANDS WARNING: Now that everyone is super overpowered and such, you could be too mighty depending on your class. Your helper mob could do too much damage and put Lichy into phase 2 prematurely, or your iLvl/weapon could affect plague damage to ghouls, etc., etc. So, be careful out there! Use weaker helpers if you can and don't be afraid to remove your gear if you need to. And don't forget that [spell=70337] scales to your level. Good luck!

Pre-Fight

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• Bring a helper mob. This mob will eventually be used to attack [npc=36597].
  
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  • This can be class spells like [spell=34433] or a Hunter pet.
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  • Or it's possible to use a trinket that summons a mob to fight for you, like [item=24124], though it will take quite a bit of patience if it's a proc. Just make sure it doesn't do too much damage or you'll end up in Phase 2 prematurely.
    
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    • Here's a nice list of trinkets with "summon" in the description. (FYI, [item=128318] doesn't work in raids.)
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• You can also set [npc=36597] as your focus to more easily keep an eye on his buffs, which can be helpful (as explained later).
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• I also happen to use TidyPlates and made sure debuff auras were turned on to help me more easily keep track of [spell=70337]. Not necessary, but helpful!
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• DBM will also tell you when [spell=70337] jumps, which is handy.
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FIGHT-O!

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2. Start the fight. DO NOT attack [npc=36597].
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4. Stand on the bottom few steps of the staircase, with your back to the throne. (This will prevent you from flying over the edge to your death from the shadow traps.)
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6. Let [npc=36597] summon some [npc=37695]s. Wait until you have at least three summons-worth (9+ ghouls) hanging around.
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8. Once you have enough, have your helper mob attack [npc=36597].
   
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   • If this helper is from a trinket with a proc you can't control, attack [npc=37698] to trigger the proc. DO NOT attack [npc=36597] or [npc=37695]s.
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10. Once your helper has aggro and [npc=36597] is attacking it, he will cast [spell=70337] on you.
    
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    • Make sure you're targettable or he won't cast it! Don't fake your own death or disappear into thin air or anything like that.
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    • Stop all damage once you're plagued. So if your helper mob is something like a Hunter pet, change it to passive at this point.
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12. Now hang out until [spell=70337] gets to 30 stacks.
    
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    • It jumps between mobs (including you) every 15 seconds. The stack count increases with every jump. After it jumps from a ghoul that ghoul will die, so don't attack them yourself and let [npc=36597] summon them all he wants.
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    • You will have to tab around the mobs to keep track how many stacks it has, which can get a bit tricky. If you have an addon that helps, great, but you can also keep an eye on [npc=36597], whom you should have focused before the fight. It's likely that his stack of [spell=74074] will be similar to the [spell=70337] stack count. It's usually one or two behind (or can be way off) so don't rely on it, but it may let you know when you're getting close.
      
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      • The plague debuff can be especially hard to find if you have a lot of ghouls. Be patient! If you're really sure that it's completely gone (either because you ran out of ghouls or you moved, etc.) then you can start over from Step 3 on the same attempt. No need to restart the fight. (Just make sure it's really gone! If you start a second [spell=70337], there's a chance you could end up without the achievement because math.)
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    • By the way, [spell=70541] can get nasty, especially if the plague jumps to you again. Try to keep yourself above 90% health while you wait.
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14. After you're sure there are at least 30 stacks of [spell=70337] (more than 30 is fine), burn down [npc=36597]. You can stop worrying about [spell=70337] now.
    
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    • Should be relatively simple to ignore the other mobs/phases/mechanics at this point if you're level 100+. Just stay on the stairs and focus DPS on him. You should be fine.
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    • If you're having trouble killing him without dying once you have the requisite number of [spell=70337] stacks, try the kiting method mentioned by liquefry to avoid plague damage. This method did not work for me -- I had to stay still -- but it may work for you if you're having trouble.
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16. He'll go on his 1:30+ roleplay rant and then you can get that final killing blow.
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18. Achievement's all yours!
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Hope this helps!

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Edits:
- Increased ghoul start number from 6+ to 9+ per a few replies saying 6 was too few, just in case!
- Added link to liquefry's reply for squishy folks who die from plague/infest damage before they can kill The Lich King.
- Added a Legion warning for higher level folks, brought to you by some helpful heroes in the replies below.
- There are now several comments that people were trying Touch of the Void as the summon trinket but it doesn't work in raids, so I added above as a preemptive warning to hopefully save people some frustration.

How to solo this achievement post-Shadowlands expansion:

Do a little hippity hop around him to aggro him.
Continue hippity hopping without hitting him.
Eventually you'll get hit with a Necrotic Plague buff.
Then it'll disappear.
Then it'll come back with a stack number.
Then it'll disappear.
When it finally comes back with a stack number equal to or greater than 30 (I waited till 34), kill the Lich King in one fell swoop.
Wait till he's finished with his theatrics.
Finish him off.
Boom! You have the achievement.

As a Hunter (I'm 566 at this time), this is extremely easy to get.

Put your pet on stay and passive, pretty much by LK, and Misdirect some aggro onto it. Then Feign Death a little bit away from them. Remain "dead" until maybe 3 Shambling Horrors are up, or 4 if you want to be safe. Once you have enough adds up, run back up and stack on them until you get [spell=70337] and let it expire off of you. Disengage backwards, Feign Death again, stay down and just tab around until you see 30 stacks on one add. I personally let it go for a bit more just to ensure the achievement ticks it off. Once you have your 30+, nuke LK and there's your achievement.

The damage from both the adds and the LK is extremely minimal for a tenacity pet with Spirit Bond. [spell=70337] doesn't seem to jump to it either so you won't have to worry about that killing it. The same goes for any other debuff LK casts.