Relevant Talents used:

Body and Soul
Solace and Insanity (FDCL works better for phase 1)
Desperate Prayer
Divine Insight
Divine Star

iLvl: 547

Lich King 25 Heroic will definitely require BiS T15, due to the DPS required more than anything else. So far the only players who have soloed this fight are those from Asian servers where they have access to higher ilvls compared to those in the Western servers.

Phase 1:
Burn as much as possible in the first 45sec to 1min (aim to burn 10% or at least close to it). At some point after the second wave of ghouls are out, use shadowfiend+fade on LK to force Necrotic Plague. The plague ticks for 150k every 5 seconds which isn't too bad but VE can be used if required. Once the plague expires, it will jump to the nearest ghoul/shambling and deals increased periodic damage. This is the only viable method to kill Shambling Horrors. The shamblings can deal a great amount of damage, especially when on low health and enraged. So it is best to start kiting the boss when more than one are out. LK also uses infest which deals increased periodic damage per tick until the player is healed to at least 90% health. Generally the best heals to deal with this are Devouring Plague and Divine Star (or others that can be used while kiting). Flash Heal is not as good since it requires hard casting (standing still) and it may not overcome the combined damage of the boss + adds. Remember there is always Desperate Prayer as last resort.
Aside from infest, the only real cause wipes in phase 1 is when the Necrotic Plague falls off completely. This occurs when a ghoul dies from the plague but there are no other ghouls in range for the plague to jump to. Also keep in mind that the plague can also jump back to the player if he/she happens to be in range and the closest to a ghoul that dies from the plague.
Tip: SW:P the ghouls to generate more Divine Insight procs for more Mind Blasts on the boss. This not only increases DPS but also healing from the increased number of Devouring Plagues.
Aim to finish this phase in 3 minutes.

First Transition Phase:
There may still be ghouls with Necrotic Plague so it's advised to keep kiting them until they all die off. During this phase 3 Raging Spirits will spawn (1 at a time at a certain interval). They have a 5 second silence effect that they cast quite frequently. If RNG is good, hopefully they all miss. Eating too many silences can lead to a wipe. Ice spheres should be kited to avoid instant wipe, or line them up in a straight line and kill them with a Divine Star. They can be ignored if the player stands at the bottom of the stairs to the throne, but getting hit by them does interrupt all spellcasts.
Tip: Try to enter this phase with 3 shadow orbs to get a head start on the the first Raging Spirit.
Aim to kill at least 2 Raging Spirits before this phase ends.

Phase 2:
Finish off the remaining Raging Spirit asap, use VE if needed. The boss retains his Infest ability from phase, as well as gaining Defile, Summon Valkyr and Soul Reaper. Summon Valkyr is completely useless when soloing, and Defile is easily dealt with. Soul Reaper increases boss haste for 5 seconds after the debuff expires so either kite him during the increased haste duration (with PW:S or feathers) or save Divine Star and/or Devouring Plague. There is always Desperate Prayer for occasions where Infest is happening at around the same time.
Aim to finish this phase in 4 minutes.

Second Transition Phase:
Same as first transition phase except that there are now 4 Raging Spirits spawning in the span of the phase instead of 3. So DPS cooldowns should be used in this phase. On a positive note, there are no longer any ghouls remaining so there is no need to worry about Necrotic Plagues anymore. Aside from that the second transition phase should be dealt with in the same way as the first one.
Aim to kill 3 Raging Spirits before this phase ends.

Phase 3:
Vile Spirits and Harvest Soul are the two new mechanics in this phase. Summon Valkyr and Infest are no longer used so health no longer needs to be topped up constantly. Soul Reaper and Defile are still used though. Vile Spirits will fixate on the player and cause an explosion once it reaches its target. They will only fixate for a certain distance so it is possible to kite them without taking any damage. However due to the immense DPS requirement of the encounter, it may not be feasible to be continuously kiting. Every 45 seconds, LK will use Harvest Soul that will port the player to within the Frostmourne for 40 seconds, during which the player cannot attack the boss. This effectively cuts the remaining time on the enrage timer by half. Aside from that its a matter of maximizing DPS on the boss at this point.
Tip: Refresh all dots, including Dev. Plague, before being ported into Frostmourne. Whilst inside, Mind Blast 3 adds so that you'll have 3 orbs upon exiting.