For this achievement, you need to play the 'All Heroes' version of Retribution eight times, as a different character each time. To access this mode, go to the Arcade, then click on 'Co-Op Mission Archives', then 'Co-Op Retribution (All Heroes)'.
I recommend playing on Normal, since the rounds will be over much more quickly, and there are no other achievements tied to the 'All Heroes' mode (so you don't need to play on any other difficulties).
Unfortunately, this does not stack with the normal Retribution mode, so you can't just play as those four characters in that mode and then a different four here. You have to play through this eight individual times.
That said, the mission is short and sweet, and it can be a lot of fun (depending on the character you choose). As always, a well-rounded team comp is highly advisable. (At the very least, you'll need at least one character who can heal someone besides themselves.) Here are the heroes I played as, plus a few notes about each:
1. Mei - I thought she was great. Since the most recent patch lets her freeze multiple targets at once with her primary fire, she's almost always able to stop at least a few enemies dead in their tracks, and her ult is great for those big fat baddies.
2. Lucio - He was a good balance between healing and damage, and his ult kept my team safe during the heavy firefights. I highly recommend him if you're comfortable with the character.
3. Winston - I thought this would be a good choice since he could damage multiple enemies at once with his gun, but I felt like I was just tickling the bad guys instead of harming them. I would not recommend him.
4. Torbjorn - This worked out a lot better than I thought it would. I just plopped down a turret during each firefight, leveled it up when I could, and spammed armor all over the place. I saved my ults for the big fatty and the last section, but I was playing pretty conservatively so I bet you could probably drop them a lot more frequently.
5. Doomfist - Boy was this one a mistake. Like Winston, I felt like I was barely damaging anyone. Even my charges only seemed to do about half damage. If you want a mobile character, I'd recommend Genji instead.
6. Zenyatta - This was a good choice. I ended the match with the most eliminations and nobody on my team died because I was able to heal-and-forget with my Harmony orb. I'd definitely recommend this omnic if you like his playstyle.
7. Brigitte - She was both effective and a ton of fun. Like Zen, I ended the match with the most eliminations because I simply hit EVERYONE when I swung my mace, and she's a strong enough healer to keep your team alive on Normal mode. Highly recommended.
8. Pharah - Very effective if you know how to stay airborne. Her rockets have a high enough splash damage that you're pretty much guaranteed to hit an enemy if you're firing in their general direction, and her ult is strong enough that you can take a fatty down by more than half health if you hit him with the full load.
Others in the comments have suggested Soldier 76, D.Va, and Mercy, and advised staying away from Junkrat (he bats enemies around the battlefield so much it makes it harder for your teammates to focus fire). If anyone else has any suggestions, please feel free to share!
Anyway, I hope this helps! I know this isn't a complete walkthrough of the mission, but on Normal it is very straightforward and only lasts about 10-15 minutes, so you should be able to pound this out in a couple hours. If you're having troubles and need someone to team up with, feel free to hit me up :)