Ok, I'm not the first one to solo Onyxia, but I tried and made it with my priest in shadow! It took me a few tries, but what a nice feeling when she were lying dead in front of my feet. I expected to get 119 gold, but only managed to loot 57g 15s 27c (the gold was grey so perhaps it was because I kicked the member who I used to create a raid in order to get into the instance in the first place).
Anyhow, gold is not what I need and I killed her for my personal achievement and this is more or less how I did it:
My secondary spec is shadow, but I changed it a bit to get Improved Vampiric Embrace istead of Shadow Reach. I used self-buffs, food, and a flask.
I initiated the fight with a Power Word: Shield on myself, a Mind Blast, Vampiric Embrace, and Devouring Plague on Onyxia before I turned her around having my back against the far wall.
Then I put a Vampiric Touch on her followed by another Mindblast to get the Replenishment ticking. After that I channeled Mind Flay and then it was time for Shadow Word: Pain.
What needs to be done for the rest of the fight when Onyxia is on the ground, is to keep everything running on her all the time. Vampiric Embrace, Replenishment, Devouring Plague, Shadow Word: Pain, and then spam Mind Flay. It is also fairly important to renew your Inner Fire all the time. There is no need to Power Word: Shield yourself if you don't get a bit low on HP.
In the second phase, when she takes of into the air and you get whelps, I made sure to refresh all the dots and replenishments before killing the adds. Shielded myself and channeled Mind Sear until there were only one left that I just nuked down.
I got Onyxia to land again after the second wave of whelps and then it was back to the same rotation as I mentioned above. You take a bit more damage here it seems since Onyxia cast Bellowing Roar so you can not heal yourself as much as in the first phase. This can be a good occasion to use Dispersion to regain a bit of mana so replenishment becomes less of an issue.
If you decide to try it out, I wish you the best of luck! It was a bit of a challenge, but a really fun thing to do!