What this mission actually requires is for you to upgrade all 3 stats (Health, Main Damage and Crit. Damage) of each class to level 2 for it to count. This is done via special currency that you can get from the shop, from the Shard shop, or the Skirmish shop/Events shop.

In other words, there are 7 upgradeable classes:

1 Fighter
2 Damager
3 Heavyweight
4 Shooter
5 Shotgunner
6 Sniper
7 Support

Each class has 3 stats (Health, Main Damage, and Crit. Damage). The mission will be complete once 5 of the classes match the requirement of having all of the 3 stats upgraded to (at least) level 2.

You can open the classes window by tapping on this clover-like icon in the team selection menu: