I recently had a customer bring in an Applie iPhone 5c with not only a bad screen but also, a disabled device. I am going to describe how to perform a reset using recovery mode. You can possibly gain access to your iPhone 5c again with these instructions.
You will need either a personal computer running most recent version of iTunes or a MAC running most current version of iTunes. Open up iTunes and then connect your USB cable to your iPhone 5c and computer.If you have an iCloud locked iPhone, this won't fix that connundrum.
First you want to hold down a top wake/sleep power button and home button at same time. Keep holding down both buttons even while you see the Apple logo. You will eventually see an iTunes logo with an arrow pointing from a cable graphic.
This means that iTunes has detected your Apple iPhone 5c. A popup window will prompt you to either update your device or restore your device. Also, iTunes will show a screen with your iPhone devices' serial number and "iPhone Recovery Mode".
You can select "Update" but most likely will need to select "Restore". Caveat Emptor, the Restore option will wipe all data off of your iPhone 5c. Update will just update the version of iOS running on your iPhone.
You might be able to access your iPhone once again by choosing either option. If your Apple iPhone 5c is iCloud locked then this solution won't work. You would need to take your device to Apple and provide proof of ownership.
This tutorial might be productive if for whatever reason you want to restore your Apple iPhone 5c back to factory. Perhaps an update fucked up your device, or it is not working smoothly anymore. At least you know how to restore your device back to factory.
Also, you learned how to update your Apple iPhone 5c via iTunes.