Credit to Raegwyn of Zuluhed EU for the awesome strategy
Credit to Daviesh for the map image.
Begin the fight by walking to Point A on the diagram. Kael'thas will start his RP. Quickly run to the bottom left exit of the room (near Point C), and run all the way back to the beginning of the instance (Point B).
Thaladred the Darkener will make his way to you. Tank him in a corner because he has a move called Psychic Blow, which is a knockback.
Lord Sanguinar will come up next. Bandage yourself in between Thaladred and Sanguinar if you have time. Lord Sanguinar's only real ability to watch for is his fear.
After you kill Lord Sanguinar, quickly run to Point C. Grand Astromancer Capernian will start attacking you along the way, just avoid her. When you get to Point C, duck into one of the corners of the room, LoSing her. Make sure you are against a wall but still in melee range. Her Arcane Burst has a knockback plus slow, so it can be very hard to get back to her if you get knocked away. She also uses Conflagration which will incapacitate you for 10 seconds, allowing her to wail away at you. Use class abilities to prevent or escape it (Raegwyn pre-emptively AMS'd it).
After you kill Grand Astromancer Capernian, proceed post-haste to Point D. Master Engineer Telonicus will engage you. His only ability of note is Remote Toy, a debuff which periodically stuns you for a few seconds. It lasts for one minute and is extremely annoying, but probably won't kill you.
Quickly run back to Point C to intercept the weapons that spawn. Use AoE attacks to take them down and spam self-healing abilities to keep yourself alive. Use DPS cooldowns here (and survivability cooldowns if needed). There will likely be 1 or 2 weapons still alive when Phase 3 begins.
No running around. This time, the mobs run to YOU. There will be 1 or 2 weapons up from the previous phase (unless you're amaaazzzing or something), but don't worry about it (especially if it is the shield, it is weak).
Grand Astromancer Capernian will be the first to come to you. Master Engineer Telonicus will come next. Lord Sanguinar will come third. Lastly, Thaladred the Darkener will make his appearance.
Finish them off the same way you did in Phase 1.
The hardest part of this phase is the Mind Control. Because you are soloing the fight, the Mind Control will not activate UNLESS there is another friendly target alive in the raid zone. You cannot have a Shadowfiend, a Ghoul, a Hunter pet, a Warlock pet, a Water Elemental, the Guardian of Ancient Kings, or EVEN BLOODWORMS active or he will reset to 100% hp when he Mind Controls you, ending combat and wasting a huge amount of time.
Quickly run to Point E, LoSing Kael'thas using the stairs.
Kael'thas has several abilities. For his standard attacks, he alternates between melee swings and chain-casting Fireball. Fireball is a slow cast which is interruptable. If you are a class with multiple interrupts, feel free to use interrupts (Mind Freeze) on the Fireballs to lessen the damage taken, but always have a spare interrupt available for Pyroblast (Strangulate).
Pyroblast is a move he casts every 60 seconds, usually right after Shock Barrier (which is similar to the Twin's Shield that the Val'kyr Twins use in ToC). Use any damage reduction cooldown to reduce the impact of the first Pyroblast (besides Guardian of Ancient Kings, obviously). From then on, try to eliminate the Shock Barriers before he finishes casting, so you can interrupt the Pyroblast.
The last part of Phase 4 is the Phoenix. Kael'thas will summon Phoenixes periodically. They do a medium amount of damage and should be killed, but priority remains on interrupt Kael'thas' casts over DPSing it. Try to use DoTs until it goes into egg form. It will also slowly kill itself. After they die, they go into egg form, in which they will be reborn if you do not DPS it down fast enough.
At 50%, Kael'thas will run away and -attempt- to go back to his room. Take this time to bandage if you want. He will stop at Point C and summon you (after a minute or so). Ignore him and air-swim/run all the way back to Point E, LoSing him the same way you did in Phase 4. Continue to DPS him as advised in Phase 4.
When he casts Gravity Lapse a second time, remain at a distance from him until the effect ends. Do not at any cost go within 8 yards of Kael'thas until he has stopped channeling Gravity Lapse. If you do, you will get tons of stacks of Nether Vapor and die, painfully.
Rinse and repeat for great success!