This isn't actually unlocked for completing your first level, but rather your first mission. When you pause the game you will see 3 missions at the top of your screen and the details required to beat them.
As far as I can tell they will randomly cycle through the available missions each day, so there's no telling what missions you will get on which days; and I do believe you can get repeat missions that you have already completed.
They can be something as simple as "Beat Duke Spookem" which just requires you to beat the First dungeon's main boss. Other times it might be for killing 500 enemies in a single dungeon.