To level up your character you need to make use of Training Droids, these can either be purchases in the 'Shipments' section, or won as you complete battles.
Each training droid has a level, and the strength of it's effect on a character depends on both the characters current level and the level of the training droid.
At the start of the game you'll only encounter level 1 or level 2 training droids, but as you progress, higher level droids become available.
You can access the training menu by going to the main menu, selecting characters, then selecting training. Then you just select the character you want to upgrade/train, then click the training droid icon. The game will automatically assign the number of training droids to your character to max him our (assuming you have enough droids).
Training also costs game credits, the higher the level the more training credits are needed.