This trophy is super easy to get in Story mode. On the first level, just play normally, except don't kill the AI AT ALL. If you can avoid getting killed at all, great. If not, wait until they kill you, and then kill them with whichever level is easiest to kill with with your character. Big Daddy has probably one of the most broken Level 2's in the game, and by the time the easy AI gets their level one you should be at level 2, and can then get an easy kill to make it even again
Just make sure the kill count is even between both of you. You never want to be ahead in kills. Just be careful not to get killed in the last 10 or so seconds as you might not get a kill fast enough depending on what level you have and your character. The first time you'll go into Overtime after the standard timer runs out. Attempt to stay away from them as much as you can, or don't let them build up meter. Then when the new timer runs out, you go into Overtime 2x. Same rules apply as Overtime. Then it'll go into Overtime 3x. Now you want to go berserk and kill them. A lot. Afterwards you should get a Tropy Pop.