You must complete all 43 DLC missions to unlock this trophy. A list of all missions are as follows:
Wanted: Dead!
Armory Assault
Bridging the Gap
Athena Set Up Us The Bomb
Knoxxed Out
This Bitch is Payback
This Bitch is Payback, pt. 2
Loot Larceny
Rendezvous
Code Breaker: Analysis
Scooter? But I Don't Even Know Her
Circle of Duty: New Recruit
Circle of Duty: Cadet
Circle of Duty: Private
Circle of Duty: Corporal
Circle of Duty: Sergeant
Circle of Duty: Medal of Duty
Claptrap Rescue: Lockdown Palace
Code Breaker: Time is Bullets
Core Collection
Mop Up
Super-Marcus Sweep
Local Trouble
It's Like Christmas!
Prison Break: Road Warrior
Prison Break: Over the Wall
Prison Break: Try Not to Get Shanked
Thrown for a Loop
Stain Removal
Lost Lewts
Boost the Monster
Greasemonkey
You've Got Moxxi: Roadblock
You've Got Moxxi: Moxxi's Red Light
OMG APC
Big Crimson Brother is Watching
Bugged
Drifter Lifter
Power Leech
Road Rage
You. Will. Die.
Purple Juice!
Little People, Big Experiments
The most common missable quest givers and their locations are as follows:
Lockdown Palace Claptrap- located halfway through the prison in a corner. See video below. Credit goes to Thegamerwalkthroughs.
Thirsty the Midget- located in a shack next to the only standing water in the Sunken Seas. The location of the shack is marked on the map by a dollar sign showing the Ammo Vendor adjacent to his shack (lower right hand side of map).
Radar Locater- located in a cave enclave in Road's End. Best shown in the video below. Credit goes to Dusk Till Dawn.
After completing the last of the 43 missions you have, the trophy will unlock.
Happy Hunting :)