The Hero Grades are Common, Bronze, Silver, Gold, and Platinum, with sub-grades in between. Each full Grade level unlocks a new skill for your Hero, which you can level up from the Skills menu.
In order to increase the grade of a Hero, you'll have to collect six pieces of Gear. The required Gear is specific to that Hero. To view what Gear is required to upgrade a Hero, tap on the "Heroes" button on the bottom right of the main menu, select a Hero, and then the list of items will be displayed under "Grade". Equipped items raise your stats, but can't be removed. Once you fill all six slots, you can tap Promote to achieve the next Grade. You will then need to find six new pieces of gear. Note that even though the Gear items are consumed during Promotion, you retain the benefits of those items.