You can use the easy mode from the options for reduced damage taken, doesn't matter for the achievement.
I used an adaptive shield for some protection against elemental damage.
But even in easy mode the fight is not THAT easy, since the dragons can level up during the battle. It is important to fight only one dragon at a time, then they remain on their starting level. Because of that you should not spawn Deathtrap or put down your turret if you are playing as Mechromancer or Commando until you defeated the first two dragons.
How it works: All dragons fly around the fighting area. If you shoot a dragon, it will land shortly after that. Then unload on the landed dragon. Do not damage the other dragons until the one you want to kill is dead. If it happens by accident and the wrong dragon lands do not damage it.
First you have to kill the electric dragon, since it has the ability to heal himself an the other dragons.
Then proceed to kill the fire dragon. He is invicible for a few seconds after landing while he is standing still and setting some of the grates on the floor on fire. As soon as he begins to move you can damage him.
Those two are the most difficult ones, it doesn't matter if you then attack the slag- or acid-dragon. You should be ok to spawn Deathtrap or to use your turret then. The acid dragon spawns some small dragons every time he lands. They are not a big problem, but keep it in mind.
