Chromatic Champion

17.12.2014 - for this day I can say there is no bug with this achievement.

If you want to fully solo this, you might have trouble with Alexstrasza part - you might be too slow to get there on time, before Deathwing assaults. If you're too late, he will assault Nozdormu first and you won't get credit. You need a speed boost which is buff "Carrying winds", applied to you while jumping between platforms. It stacks up to 2, so you can have 120% speed boost for 9 sec. All you need to do is to start encounter with 2 stacks of Carrying winds and rush to Alexstrasza platform.
Hope it helps.

Killing Ysera first:
You lose Ysera's Presence and her Enter the Dream special ability that reduces all your damage taken by 50% for 5 seconds (30 sec cooldown) which can be helpful for reducing damage taken from Elementium Bolt, Impale or Blistering Heat. This is the platform you will start on when you are fighting the boss normally simply because all the other dragons give you much better buffs.
The best order when starting on Ysera's platform: Ysera -> Alexstrasza -> Nozdormu -> Kalecgos

Killing Nozdormu first:
You lose Nozdormu's Presence and his Time Zone that slows the Elementium Bolt and gives you time to destroy it before it even reaches the ground and does huge damage to everyone in the raid. Time Zone also slows the attack speed of Regenerative Blood while it stands in it so the Degenerative Bite debuff it applies on the tank stacks slower. Starting on this platform shouldn't be too difficult since the buffs you gain aren't that great and you can mitigate the damage from the meteor crash with Ysera's Enter the Dream special ability.
The best order when starting on Nozdormu's platform: Nozdormu -> Alexstrasza -> Ysera -> Kalecgos

Killing Alexstrasza first:
You lose Alexstrasza's Presence and her Cauterize which helps you kill the Blistering Tentacles that spawn from Arm Tentacles. The Blistering Tentacles are immune to AoE damage so your DPS will need to be fast on killing them before damage gets too high.
The best order when starting on Alexstrasza's platform: Alexstrasza -> Ysera -> Nozdormu -> Kalecgos

Killing Kalecgos first:
You lose Kalecgos' Presence and his Spellweaving which helps you killing the Regenerative Blood that Deathwing spawns during Hemorrhage. This will be the hardest way to defeat the boss because you will need VERY high DPS to kill the Regenerative Blood on the remaining 3 platforms without Kalecgos' buff.
The best order when starting on Kalecgos' platform: Kalecgos -> Ysera -> Nozdormu -> Alexstrasza

To get to Alextraza in time:
Before you start the encounter, jump to the nearest platform then back to Thrall. That will apply two stacks of Carrying Winds which increases your run speed by 120% for a 10 seconds just enough for you to start the encounter and get to Alex in time for Deathwing to assault her. Wait for the tentacle ([npc=56471]) to spawn and impale Alex then kill it, kill the leg and continue with remaining Aspects. The order is irrelevant now since this event is soloable.

NOTE: Still works as of 2023, thanks for confirming everyone!