Follow the iPhone 5C home button flex cable replacement guide until step 8, and test the phone. If the issue stops, it means your home button flex cable is damaged or stuck somehow. May be due to water damage. Then follow the remainder of the steps to replace the home button flex cable.
If your issue remains, it means the damage if further down the line, probably at the level of the logic board. At this point you need to continue till step 13 to completely remove the screen, then examine the visible parts of the phone. You need to look for traces of liquid, corrosion, and water indicators turned from white to red. Let us know if you something of the sort.
Even if your home button flex cable is fixed, have that look. If you see traces of liquid, look for other threads about water damage repair, and follow the repair guides on this website to remove and clean the logic board and other important components. If you are not sure on how to do it, post a separate question.