While I disagree with the earlier solution's assessment that this achievement is "fairly easy", I did manage to unlock this achievement with randoms. But it was far from my first try.
However , once you've learned the specific role of each character, assuming you find teammates who have as well, beating Uprising on Hard shouldn't be nearly as difficult as it seems.
Here are general pointers for playing according to how I've learned to play:
Tracer: You need to take care of every shield user you see, and save your pulse bombs for Bastions once you get to defending the payload. For the Omnics with shields, simply dash around behind them and shoot their left arm. You should destroy it in less than a full clip, and their shield will be gone, leaving it as an easy target for your teammates. For Bastions, get their attention and continuously move around them while dashing back to cover and you should be fine. But get to them fast-- Bastions are almost always the reason we wiped. If you have smart teammates who are damage boosting you and stunning the Bastion, your job should be easier.
Reinhardt: Don't get greedy. Your shield is incredibly important, and when you go out of position, it leaves the whole team at risk of going down before you get back. Save your hammer for groups of little guys or large enemies who have gotten too close. Only charge when necessary, specifically when an exploding omnic comes toward the payload. Those things do major damage, so when they get close to exploding, simply charge them away-- but not TOO far away. Avoid going back into another area. Guide them into a wall. Charging or Ulting is also useful against Bastions that your Tracer is having trouble with. Do not charge or Ult omnics with shields by the way; Tracer should take care of them faster than you could, and you get to stay in position.
Torbjorn: Your job isn't complicated until the final push, where you'll have 5min to kill four OR-14s. The team will need to be moving forward as a group behind Reinhardt, which means you won't be able to lay a turret and stick near the point anymore. Just keep moving your turret and not even worrying about upgrading it to level two. Set it where it can have a clear shot from behind Reinhardt's shield, and shoot at every enemy you see to build up your Ult. Use your Ult liberally, as there's nothing you should save it for here. Stay near cover and watch for Bastions, and try to help your Tracer kill them from long range.
Mercy: If you're not healing, you're damage buffing. One mistake I constantly see with Mercy users is that they stick to healing the whole match, afraid that if they stop their teammates will keel over. Much of this mode requires quick kills, and damage buffing a teammate (particularly Tracer since she has some of the most important objectives) gets them out of danger faster. Use your pistol to headshot flanking enemies quickly, but don't focus on them for too long. This mode is unforgiving once a teammate goes down, so it's imperative that you don't let them. Speaking of which, when a teammate goes down, don't waste your Ult right away. That should be saved for emergencies and multiple resurrections. The smarter option when one teammate goes down is for you to heal another teammate as they pick them up with X (just to make sure they don't go down as well while trying), or pick them up yourself from behind Reinhardt's shield. Never try to pick them up yourself out in the open. You will go down, and you will wipe.
General tips: When a Bastion is on the field, he takes priority. He has to be killed. Play through the mode on Normal multiple times before attempting Hard, so you can get a feel for when waves occur and what enemies come from where. The enemy makeup doesn't change with difficulty; they just get harder to kill. Only one person needs to be on the point when you're hacking; multiple teammates don't make it go any faster. On the final push, particularly the second half where you have to kill two OR-14s at once, DON'T RUSH. Five minutes is plenty of time. You should be able to clear out the first two OR-14s within about a minute, so don't charge in thinking you have to hurry, then get downed by a Bastion. Fall back behind cover, and use the path on the left to kill the two Bastions sitting along that route. Once you've cleared the Bastions and the whole team is healed, then bumrush the last two OR-14s. A damage buff on Reinhardt plus two guns should make quick work of them before the next Bastion spawns.
Optional: You can avoid the entire moving-the-payload section by sending Tracer all the way to the end of the route (where the doors will blow up) before the payload starts to move. It somehow cancels out all enemy spawns as long as Tracer stays within a few yards of that door. Credit goes to the solution for the Legendary achievement, where I found this info. It definitely works, and can help with the achievement for clearing hard with an 80% payload health.
Also stating the obvious: This achievement can only be unlocked in standard Uprising, not the any-hero version.