You can go through the steps of changing your battery and your lightning charge port. If that helps good. But like many 5c owners the u2 ic goes bad triggering these issues. I have read it's due in part to a 3rd party sync cable. Which is odd in my case I always use the original cables but my 5c will not charge any more nor will it come on. If this is your case and you are still under warranty with apple go immediately and try to exchange it. Like so many whom are not under warranty are out of luck unless you know some whom can micro solder to replace that chip.
this is happening to me try changing out the screen, battery and charger port all at once, if that doesn't work its a bad motherboard, try to reflow the motherboard. in my case a new screen battery and charger port did not work out for me so I'm going to attempt a reflow
Okay I had a problem with the iPhone 5c and 5s. For the 5s it was completely dead..so I took it off charge for 10 minutes! When I plugged the charger in it I kept hold of the top button aswell and it came on straight away. with the 5c I kept hold of the top button for around 30 seconds and then just left it,it came on. I hope this helps you :))
