To earn this achievement that's called "Repaid in Blood", you must complete a Vendetta mission.
The way it works is simple. One of your friends, on your PSN Friend List, should be playing the game (I hear it doesn't need to be a friend, but a total stranger also works too. I can't confirm that, however.). If either him or her gets killed by a captain or a warchief, a Vendetta mission will show itself on one of the two maps (Udon or Sea Of Nurmen).
Get to the starting point of that mission and load it up. Your job is to then find the Uruk responsible for the death of your friend and avenge him or her by killing him. Once you have successfully killed that specific captain, uruk, or even a warchief (doesn't matter who really), the trophy should unlock.
I should also mention that a bonus objective would be to complete the mission within a certain time limit, but even if you don't complete the bonus objective, that should be OK. As long as you complete the main objective, you're fine.
For more details on this trophy, I've attached a visual guide. Credit goes to MrReign for the video guide on how to obtain this trophy. I hope it helps! Good luck!