Had to be done!
This is the normal version of the Lich King 10-man, but the 10-Heroic version is undoubtedly also doable, I just didn't have the raid tag for it.
I still regret not having had enough time to kill him at 85, 'cause I really think he was doable then (best try at 64%).
At 90 there's lot of ways of doing it, mine is probably one of the most tedious ones, but it's really safe. I use Dark Apotheosis with Soul Link and the 2P T5 bonus which heals your pet (=you through Soul Link) by 15% of the damage done. Add the 30% ICC buff and lots of AoE dmg, and you're basically at full health the whole fight :p
P1 used to be the hardest phase. I tank the boss myself so I don't get the Necrotic Plague. My pet tanks the Shambling Horror apart. The Aoe dmg from Immolation Aura and Harvest Life are way enough to keep me above 90% hp, so the Infest is a no issue. But Dark Apotheosis' dps is abysmal, so that phase lasts forever. At around 80% I have my pet taunt the boss so I can get the Necrotic Plague. I teleport near the Horrors and dispell the plague with a Purification Potion, that way the plague will jump enough time to kill these adds before P2.
First transition is simple, have your pet tank the Raging Spirits away from you, don't get bumped by the sphere.
During P2 you won't get picked up by the Val'kyr if you've been hit by the boss (during the last 30-60s I think). So you can have your pet tank the boss and dps in caster/Meta form, as long as you get aggro back soon enough. I didn't take much risk there.
Second transition is easy (had time to dps in Meta) and with no more Infest P3 is trivial.
With all defensive cds I managed to survive the Fury of Frostmourne "instant kill", but as you can see you can't finish him until the roleplay ends :p
Demonology Spec. Talents are Harvest Life, Mortal Coil, Soul Link, BR, Grimoire of Supremacy, Archimonde's Vengeance. Glyphs are Demon Hunting, Siphon Life, Demon Training.
409 ilvl (451 ilvl without the 2P T5 :x).

Here we are again, now in heroic difficulty!
Just for the record, all ICC bosses (minus Gunship and Dreamwalker obviously) are now soloable in Heroic 10-man.
This one's harder than on normal mode (you don't say?). The boss has now 29.5M hp so because of the enrage timer I can't use the Dark Apotheosis tanking strat that worked so well for me.
I went Affliction using Grimoire of Supremacy AND Soul Link. Because it's the spec I'm most confortable with. Where do I get all my heals from then? Well... from my faithful T5 2P bonus of course (15% of all my damage converted to heal).
Phase 1 is still the hardest. The Shadow Traps should be easy to avoid, but there's sometimes so much going on that you (= I) just simply forget... well glad I had that Demonic Teleport ;)
Necrotic Plague can be still annoying, I keep some Purification Potion just in case I get a full stacked one. This try it was not needed, and the first transition was a clean one with the remaining Shambling Horrors dying just as it started.
Phase 2 can be tricky, depending on how many adds you still have left. You need to let the Lick King hit you, or the Val'kyrs will grab you and toss you off the platform (no I'm not using that buggy Teleport tactics). When you've taken one hit from the boss, you will be seen as a tank for one minute exactly. So you have to re-summon you pet to get aggro and just have him taunt after ont hit. The only thing that can kill you is the Infest, and that one at 4:34 could have been a wipe.
2nd transition is easier than the first because you don't enter it with any add left. And there's no more Infest -so you won't need to be at full life when it ends, nor Val'kyrs.
Reaching Phase 3 should be a kill. Use cooldowns to survive the Vile Spirits if necessary.
No Undending Resolve to survive the Fury of Frostmourne this time, but you alredy know how that'd be :p
Affliction Spec. Talents are Dark Regeneration, Mortal Coil, Soul Link, BR, Grimoire of Supremacy, Kilj'aeden's Cunning. Glyphs are Soul Shards, Siphon Life, Healthstone.
432 ilvl (478 ilvl without the 2P T5 :x).