The big thing about the brawl is you have to coordinate.
Soldier - Your main focus is popping regular Zomnics, moving around the map and dropping biotic field whenever someone gets hurt. Save ultimate for Shock-Tires. Use missiles liberally to clear Zomnic groups, but try to time it so you have one for Shock-Tires (just in case you don't have ult up). Your main focus will always be the Zomnics getting close to the door, but don't hesitate to use some rockets on a boss or two.
McCree - Hug the left stairs, can hold the lane by yourself for massive ultimate charge. Save ultimate for boss fights, preferably when Ana has her ultimate ready. A boosted deadeye can one shot any of the bosses very quickly, with only the Monster taking some time due to his huge health pool. If you don't kill a boss in a single Deadeye then leave the stairs for a flashbang, FTH, flashbang combo to finish them off.
Hanzo - Stay near the door and help mostly with Zomnics. Focus on the ones that launch bombs so the other heros can focus on the swamers. Use ultimate for lane clears during most of the boss fights and stay away from the bosses themselves, but don't hesitate to save one ult just before Junkenstien comes out, as you can climb up to his perch and launch an ult that Junkenstien won't be able to avoid. Use scatter arrow liberally in doorways or at the feet of bosses if you have an opening.
Ana - Keep an eye on health and focus your main attacks for healing (but help DPS if everyone is topped off), use grenades to kill Zomnics. Make sure to also use the grenade on the Monster, since he will often attempt to heal himself. When a boss spawns, use sleep dart to knock them out, allowing you to clear the boss wave or setup the McCree/Ana ult combo. The most important to sleep is the Mercy on the last push, as she makes the Monster incredibly deadly.