Firstly, I apologize for the messy ending. There are quite a number of factors which could have prevented it, which I will explain below. Believe me, if it wasn't for the fact that I couldn't kill her for another a week, I would have definitely discarded this video and tried again. But hey, a kill's a kill.
From a shadow priest perspective (or any other hybrid caster/healer DPS for that matter), this fight is possible since ground mechanics can be completely avoided, and harsh winds debuff can be healed through due to the haste and crit buffs.
The first phase of the fight begins with Alysrazor cleaving across the middle, so stay out of her way. She will leave behind feathers which can be picked up. Pick up 3 to fly into the air. Once airborne, DPS her while following her closely. Make sure to fly through all the rings she spawns. Each ring will provide a haste buff which stacks up to 25. A nifty trick is that if you happen to be right on the ring as it spawns, and still be on it as it despawns, then you will get 2 stacks of the buff, which means 25 stacks can be reached a lot sooner, heavily boosting DPS. However this trick takes a bit of practice to perfect. I actually did not know about this trick, and it definitely would have saved me from messy ending. Along with the rings, she also spawns fiery clouds which needs to be avoided.
After around 40 seconds of flying, she will cleave across the room again, and will molt more feathers into the ground. Ignore the feathers and stay close to her or else she will bug out, but not too close. Keep in mind that she will still spawn the rings whilst molting. After that she will resume her flying and just continue to follow her and collect the rings. I personally missed a few rings in the video, and caused a bit of a DPS loss. She will then molt again, and simply repeat the same process.
After she has molted twice, the third time she will begin the fiery tornado phase (phase 2). After this phase begins, there are still around 7 or 8 seconds left on your flying ability so make use of it as much as you can before flying out of the area. Dont fly out as soon as this phase begins or else you will be spending longer time outside, causing harsh winds to stack to unhealable levels. Once outside, the harsh winds will begin to tick. Use Greater Heals to heal through this. Place a renew tick on yourself when the damage becomes 40k per tick. When damage reaches 70k to 80k per tick use dispersion. However if you did not immediately fly out when phase 2 starts, you should not be getting any 80k ticks and only a few 70k ticks at best, so dispersion is probably not even needed. Hence why other caster/healer classes can solo this as well.
Once the Harsh Winds ended, phase 3 begins. For this phase, just move up Rhyolith's ramp, but not too far up or else the boss will bug out. Once she reaches 50 energy, phase 4 starts. She will rise up and begin to cleave towards you. Hopefully you are far away enough that by the time she reaches you she will be at 100 energy so she flies back up again and phase 1 repeats.
From this point onwards just repeat everything mentioned above. However keep in mind that after the third tornado phase, she will not fly back up despite reaching 100 energy. If the boss cannot be killed by this point, then it will be a wipe. So there is in fact a DPS race involved. With that said, the reason for my messy ending was exactly because I didn't manage to kill her before tornado phase hits, since I missed a few rings. But she was only on 1.5mil so I decided to just not fly out and at that point it was all or nothing. I probably should have flown out when and let the dots finish her, and that would have been a much cleaner kill but I guess I was too nervous at the time and decided to take a gamble.
Also it is worth noting that this fight favours mastery over haste and crit. Personally I didn't spec too heavily into mastery myself at the time of this video, but if I did, I definitely would have done a lot more DPS. Now I'm just banging my head for not reforging /facedesk