Before deleting any folder or file mentioned below, it is a good idea to make a back up.

First thing to try is deleting the save data for your current run through a campaign. This can be done by deleting your current "Run" folder which can be found in the following location:
...\Documents\My Games\Popup Dungeon\Backer-Alpha 4\Profiles\Default\Game Progress\Run

If the game still won't start, the next thing to try is deleting your "Game Progress" folder found in the following location:
..\Documents\My Games\Popup Dungeon\Backer-Alpha 4\Profiles\Default\Game Progress

Last thing to try is deleting your "Default" profile data folder found in the following location:
..Documents\My Games\Popup Dungeon\Backer-Alpha 4\Profiles\Default