Classes are unlocked by clicking on the clover icon in the team selection menu. You will see all 7 of them.

Each class has 3 stats (Health, Main Damage, and Crit. Damage). These are the ones you need to upgrade using a special currency you can get from the Shop, Shard Shop, or Skirmish Shop.

In your case, you need to upgrade one of the stats of the "Shooter" class.