Raiders of the Last Boss

Walkthrough
Go to the Lair of Infinite Agony, and from there travel to the marked objective. Put 20 Eridium into the bowl, which will allow access into The Winged Storm. From here, just travel down the path, and drop down to the arena, and activate the highlighed scroll, thus beginning the fight.

It is advised instead to have a good Adaptive Shield such as Evolution or Neogenator due to the elemental attacks, and because the Vault Hunter's shields are likely to be down for a large portion of the fight. Also, a Moxxi weapon such as the Grog Nozzle is highly recommended for this fight, seeing as it is the best way to regenerate health fast enough to stay alive.

The order in which the dragons are killed does not necessarily matter, however one popular order is: Boost, Healianth, Incinerate, Brood. Boost and Healianth are taken out first because they make the fight more difficult the longer they are alive, and Boost can Slag the Vault Hunter(s) even when he is in the air, which can be very problematic, especially if Incinerate lands. Brood is taken out last because his lizards can provide easy second winds.

Before getting Incinerate to land, it is helpful to use the Defense Shrine and/or throw some Transfusion Grenades, seeing as it may not be possible to regenerate health by damaging him with your gun because of his wings blocking the bullets, and he does the most damage to the Vault Hunter's health.

Salvador seems to be the most viable character for soloing the Dragons, seeing as he can have a Miss Moxxi weapon in one hand, and a high damage weapon in the other hand, which lets him regenerate his health and do lots of damage simultaneously. Maxing out  Money Shot  and  Inconceivable, and using a class mod that gives +6 to Money Shot is invaluable  during this fight, seeing as it is best to be able to kill each Dragon (except Brood) without having to make them land twice. Also, it is not a good idea to use Filled to the Brim at all if utilizing Money Shot, because that allows the player to output a massive amount of damage in a very short time, which can kill a dragon in one landing.