This achievement is very circumstantial (maybe even more so than other Overwatch achievements) but can be obtained through statistic planning and well, mostly luck.
So, Winston's Primal Rage ability lasts 10 seconds and gives you 1000 health. You can swing his arms every second and a half, his jump cooldown is at 2 seconds now instead of 6 seconds, and you cannot use his shield or Tesla gun.
The best place to pop this achievement is on King of the Hill maps when the enemy has control of the hill. They will usually be placed in easy to guess defense positions and hopefully not moving around as much as they would be if they were trying to capture the point. Or on the contrary, if you have control of the hill, wait until they make an offensive play where they all are trying to take the point at once. They will be more inclined to want to hold their positions and hopefully you can get they all.
Your arm swings are very wide and you can jump very far, the only real issue with this is that you have to hit six enemies, which just so happens to be the exact number of people on the other team. The odds of them all being alive AND together isn't a common site to see in this chaotic shooter, so hold out and try to catch every opportunity that you can to use his ult.
Also, last thing, this achievement might either be glitched or worded weird. When I got this achievement, I am almost positive that I only hit five enemies, as their sixth was coming out of their base when I went into cooldown. It may be possible that turrets count as enemies (Symmetra turrets DEFINITELY do not count), or hitting the same enemy more than once counts. If this is the case, I would wait for a room with Symmetra turrets, and just go to town destroying them all, and then looking for more enemies to hit just to be safe. Get people in small rooms when you run your ult and bounce them off the walls and hit them as much as you can. We will need more confirmation and proof for this theory, but just jeep it in mind if you know you won't be able to hit all six people on the opposing team.
If you have any suggestions or comments, let me know and I can always come back and edit this. Also, if you got this achievement in a weird way as well, leave me a comment so that we can narrow down the possible ways to get this, thanks!