This achievement is story based and should be completed during natural progression. Hunting the Firehawk is a main mission that chronologically occurs as below:
My First Gun
Blindsided
Cleaning Up the Berg
Best Minion Ever
The Road to Sanctuary
Plan B
Hunting the Firehawk
Essentially, picking up the message from Roland that completed the previous mission will start the new quest. He instructs you to open (and introduce you to) the Safe which is in the back room upstairs and take the ECHO Recorder. Scooter pipes in after the recorder plays and instructs you to get to Frostburn Canyon to reach The Firehawk and rescue Roland. Go SW to leave Sanctuary and enter Three Horns Divide.
As soon as you enter the area you'll be next to a Catch-a-Ride on your left. Drive/walk/fight your way S to enter Marrowfield to the E near another Catch-a-Ride. Fight through this small outpost to get to the entrance for Frostburn Canyon.
The Firehawk can be found in the NW but a cliff prevents you from taking the most direct route. Thankfully, there are 7 Bloodshot signs (banners with Psychos nailed to them) literally pointing the way. The best approach is going NE through Blisterpus Camp, S into Blacktoe Cavern, W over bridge near Frozen Ant Lake, N through Ashmouth Camp and finally N into Firehawk Lair.
You'll be introduced by the sound of distant earthquakes and Handsome Jack saying his drill is nearing the Vault. Load up at the Ammo Dump left of the lair's entrance then make your way to the platform to trigger the cutscene and find Lilith (aka The Firehawk). Turns out, she's on your side and has been fighting bandits so Roland can pursue Jack. Revive her and bring her Eridium from a strong box to the N. She informs you that Roland was captured shortly before bandits start pouring onto the platform.
Fight off all the waves and grab all the loot scattered about. Then head to the platform to the W and take the Eridium nugget in the box back to Lilith. You can then turn in the quest to her.
The achievement should pop and you'll be rewarded with:
- Approx $22
- Approx 1187 XP
- Class Mod (White)