Piggybacking off of the other guides, I honestly think Lucio is the best pick for this achievement. His healing song heals all allies around you without you needing to actively do anything other than just being near them.
Combine this with objectives like pushing the payload, where your team will be constantly taking a lot of fire from the enemy and the healing points add up, especially when using his "Amp It Up" ability to increase it's effectiveness in team battles. As long as you make sure to keep around the majority of your team, and you don't have another healer on your own team, you should definitely get the healing postgame card at the end of the game.