To get Vulak`Aerr to spawn:
1) Make sure that Thylex of Veeshan is alive.
2a) If [Aaryonar AND Lady Mirenilla AND Lady Nevederia AND Lord Feshlak AND Lord Kreizenn AND Lord Vyemm] are already dead then wait for one of them to spawn.
2b) If [Aaryonar OR Lady Mirenilla OR Lady Nevederia OR Lord Feshlak OR Lord Kreizenn OR Lord Vyemm] is up then kill all that are up.
3) Upon the death of the last dragon, Vulak'Aerr spawns.
Note: Lord Koi`Doken does not matter and can be dead or alive.
This was updated with what appears to be previously undocumented information: Aaryonar also triggers Vulak'Aerr. This was tested and confirmed on a live server. Aaryonar is unique in that it will agro from across the zone and quickly chase down any character fighting many named Lords and Lady's in the north wing of the Temple of Veeshan. If you happen to start clearing the zone far and away from Aaryonar, you'll agro and kill Aaryonar anyway. In the confirmed test case today, all other Lady's and Lords were dead before Thylex of Veeshan spawned, then Thylex of Veeshan spawned and minutes later Aaryonar spawned. Killing Aaryonar alone with Thylex of Veeshan alive triggered Vulak'Aerr to spawn.