The iPhone charging subsystem can be damaged by non-MFi (Made for iDevice) certified chargers. Based on the sequence of events you describe, this is what happened to your phone. Cheap dollar-store charging bricks also can cause this.
While it is possible that your battery is defective, I believe that the Tristar(U2) IC on your device was damaged by the USB port. This is not a DIY repair unless you have experience in micro-soldering.
If you have access to a known-good battery, you can always replace it. If a known-good battery does not charge, then it is a U2 problem.
iPhone 5c Battery Replacement
The 5C also suffers from micro-fractures on the logic board so this could also be the root problem on your device. A qualified repair shop that does micro-soldering should be able to troubleshoot your device properly if you don't want to do it yourself.