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Rookie & Showstopper: Resign this save, open your PS4 profile you want to earn the trophies on, & copy the save over. Once you do turn on a second controller & log into any account other than the one you're trying to get these trophies for. Even a guest account will be fine. Anyways, open up Doom, select "Local Multiplayer", "Deathmatch". I chose the map E1M1: Hanger map, it's under the "Knee-Deep in the Dead" chapter, it's the very first level of the game. I went to frag limit, then time limit & chose whatever I felt comfortable with, max frag limit 25 is the max, x4 of those matches make 100 kills. Press R2 to start the game, find your friend & kill him, the game will lag once you do, but both trophies will definitely pop. You don't even need to end the match, just kill him.

If the second account you're using wants to earn these trophies without the legitimate gameplay he'll have to resign this save to his profile & follow the steps above, once he does that he can earn them. They will not earn the trophies just by you being the "host" of the game.

SIDE NOTE: If your account that you're trying to earn trophies on is not activated as primary, then the game will appear locked to all other users. This trick allows for numerous accounts on your PS4 to play the game. I do believe they can join your game regardless though if it is activated or not.