To unlock the Light Machine Gun Lover trophy in Anthem, you will need to complete three of the Light Machine Gun challenges which can be found by opening the Menu > Challenges > Weapons > Light Machine Gun
You will find four challenges available -- one for each of the game's three LMGs, and a final one for completing all three of the LMG challenges.
The three challenges you will need to complete are 50 kills with the following three weapons:
Relentless
Havoc
Sledgehammer