You unlock Boss Survival after you complete the game (tanker and plant) once. This is located in the "Special" option under the main menu.
You can pick the difficulty and the character you want to use for this. If you want to just burn through this quickly, pick Raiden so you have the HF Blade during the Solidus fight.
Boss Survival is just that - surviving boss battles and/or waves of enemies. Once you beat one round, you advance to the next. You are timed, but it really doesn't matter - just beating every round will net you this trophy. The difficulty doesn't matter either - you can play this on easy and still get the trophy.
Also, keep an eye out for items and ammo - most of the time you will find stuff during one fight that you can use in the next one.
Questions, corrections and neg vote justifications below.