Middle-earth: Shadow of Mordor

Vendetta missions are denoted by red shields on the map containing a circle with what looks like a V.
Vendetta missions task you with killing a captain that has killed another player. I believe if you have friends that have died they will be the first offerings of these missions. About the first 10 I did were avenging friends and then I got randoms. (Also credit to xGamBeaTx for confirming.)
They involve searching a green circle on the map for the captain, and then killing them however you see fit (usually involving some bonus objective based on their weakness.)
When you find the Uruk, it automatically tells you all their stats, so there's no need to find info on them yourself.
Once you've killed the Uruk, the achievement will unlock.

So I made a video guide on how to obtain this achievement (has commentary on how I did this). As said in above Solution, you don't need friends that play this game for a Vendetta to appear. I had a few friends ones appear in the first area, but once I unlocked the second area I noticed I was getting Randoms. Also be aware that when doing this I also noticed (When I first attempted this) there is two other captains around, no where near your target, but once you attack your target they join in and make it a little difficult. What I did was searched for them first and took them out before heading to my target. As shown in the video.

Up to date solution as of January 2021

While this achievement was scheduled to be discontinued when Vendetta missions were shut down at the end of 2020, it has now been patched and will remain obtainable.

Per the developer:
Captains that have killed you before are marked in the Sauron's Army screen as a Revenge Target with a red triangle over their head. Kill one of them and the achievement should unlock.

I had a little trouble getting this to show up, and since the clock is ticking, figured I'd share my experience.
Having your account email verified on WBPlay doesn't seem to be necessary. I'm gonna present two options here, the one I used and an abbreviated one since I probably did some unnecessary stuff.

My procedure:
1) Sign up for a WBPlay account, email verification not required.
2) Start the game and play through the first 3 story missions: The Slaver; The Spirit of Mordor; An Interested Party. At this point red side missions should start showing up like the ones I mentioned above.
3) Switch to a second account, sign up a for a WPlay account with them too.
4) Start the game and play through the 1st 3 story missions. Find a captain and let them kill you. The captain that showed up in the mission was actually one that killed me before finishing those 3 missions, so it might not be strictly necessary.
5) Quit and log back in with your primary account
6) Advance Time several times? No luck
7) Start heading towards a random uruk to interrogate him? Jackpot, before the guy even saw me.

Quick procedure.
1) Sign up for a WBPlay account with your secondary account
2) Start playing with the second account and get far enough in the story that you can be killed by a random, non-story captain. You certainly need to complete the tutorial, may need to complete "The Slaver" also.
3) Switch to you main account, sign up for WBPlay.
4) Play though the first 3 missions.
5) Wander around until the vendetta mission pops up. It may not show up while you are in a story mission, it may not show up in a side mission. Mine showed up while traveling out of missions. Took less than five minutes after I stopped just advancing time.