In order to get the trophy that's called "Under The Red Hood", you will need to complete the DLC story for Red Hood from start to finish. You can either get it through the season pass or, if you don't have the season pass, you'll have to pay for it on its own.
When the game starts, Red Hood will tackle on a bunch of Black Mask's thugs in a huge fight. If you're pretty good with the combat system, it won't be much of a problem. If not, I suggest evading all the time, using counters when appropriate, strike when you can, and use certain takedowns. If a guy is charging at you, press
to take out your guns and eliminate him. When it comes to guys with the electric stick, you can jump over them and hit them with behind, use your gun, or do a special combat takedown.
After that fight, it is now time for a predator situation. This is unique, because there are guys in the room that wear body armor, so guns would be useless against them. Start off with the sniper on the right by doing a silent takedown (by pressing
when you are behind him). Once that's done, play it safe with the guys in body armor by doing silent takedowns on them, then eliminate the thugs without body armor in anyway you like, just as long as you don't die.
Finally, it's the final fight of the DLC. By now, the thugs are pretty standard, so just destroy the first batch. The second batch comes out, this time with Black Mask leading them. Black Mask will eventually take out his gun and shoot Red Hood. You can counter the move with the button prompts that appear on the screen and he will get his by Red Hood's gun. You can slowly defeat him that way or you can a perform a beatdown and hurt him that way. The thugs will keep on reappearing, even after you defeated some of them, so split your time between defending yourself against them and hurting Black Mask.
Once you empty the energy meter for Black Mask, the DLC ends and the trophy is unlocked. If you need a visual aid on how to complete the story and unlock this trophy, I have provided a video guide. All thanks goes to Batman Arkham Videos for playing through the DLC and creating a video to show the results.
Happy hunting and good luck!