A Friend of mine had the some problem with her iPhone 5c and we worked out that it was the charger,.
try these steps to try and resolve to problem.
1, Check the charging port of lint
2, If changing via USB on a computer then try using a mains charger
3. If charging via a mains charger then try charging from a computers USB port
4. If option 2 or 3 didn't work then try options 2 & 3 again but with a different cable
It could also be that the phone a froze, if that is the case then try doing another soft reset
If none of the above work then I suspect either a dead battery or a hardware fault.
If all else fails then book an appointment at your nearest apple retail store and take your phone, cable & charger in for testing.
If your phone is less than 3 months (1yr for uk) then the battery is covered under the warranty.
also have you checked with apple as I do know that a small number of iPhone 5's were shipped with faulty batteries
https://www.apple.com/uk/support/iphone5-battery/