I accidently overcharged my iPhone 5c and it stopped working. What should I do?
Theoretically overcharging a device is not possible, as once it reaches the 100% of the battery capacity, it will just stop charging. 

Are you sure the device was being charged when you left it? Perhaps you connected the iPhone to the charger but not the charger to the AC? 

This will explain why it won't turn on as it would be battery drained.

If this is not the case, going to an Apple Store would be the best option.

It's far more likely you just had a failure, unrelated to charging and need service. If it was overcharging, getting a technician to look at it would be even more important that a simple failure, so have it checked out.