The cable that plugs into the 5c port does not always work.
Compressed air, a wooden/plastic toothpick and a flashlight.
Use the flashlight to check for a build up of cruft inside the port. If you see any, use either the toothpick to pick it out, or the compressed air to blow it out.
If that doesn't solve the issue, then do as D suggested and try a different cable.
If all else fails, make an appointment with an Apple Genius bar.