To MiiHere's point try a different charging cable / wall plug. If the battery is truly dead you can have it replaced / swapped out at Apple. There is a fee for a battery swap. Also, 3rd party repair shops can swap out the battery as well. iFixIt offers a breakdown of the steps for anyone that is wanting to do the process themselves.
5c specific - https://www.ifixit.com/Device/iPhone_5c