I did this with a random person in public matchmaking but it wasn't entirely too hard to time. Basically when an enemy Ooze is staggered by gunfire, the A prompt to charge your character's unique melee attack to hit it with pops up when you near it. I waited for my random teammate to walk over to it and begin the stance for the fully charged attack to which I followed suit. The achievement unlocks if you both hit the enemy with a fully charged attack, it does NOT need to be at the same second. Your hit just has to land on the enemy. Both of your fully charged attacks need to hit. Though essentially you need to be pretty close to the enemy and at about the same time frame for you both to strike and hit the unfortunate Ooze.
For boosting partners or a friend, I'd suggest it's easiest if you play with mics and signal to your partner on the mic when an enemy is staggered so you can initiate the charged attack.
P.S. This was not meant to be so long winded.
Best of luck!