Bottom right of the display you should see three icons. Each one is a goal. The numbers on these icons represents how many times you have to achieve each goal. Goals can be killing types of monsters, using a specific damage type and looting objects.
It is likely not worth paying much attention to until you've been playing a while anyway. However, when you hit all of the goals you will get some loot that is your level. So one way to clear the mini game is go back and clear lower level instances for fast kills.
My main is set up with mods in the weapons for each of the damage types just to help the mini game complete faster.