Iphone 5c gets really hot

Could be a battery issue. I would get the phone checked out just to make sure. There have been reports of some phones (not I phone) catching fire due to faulty battery issue.

I just replaced an iPhone 5 that had the same issue. And the battery life was terrible. My suspicion is that iPhone 5 doesn't run the latest iOS efficiently.

my 5s had same thing

got $400 trade in to new 7 with current verizon promotion(yesterday)

credit is monthly ($400 / 24 mos) and nets against new phone monthly payment

Check to see if you have apps running in the background. Go into settings and disable background app refresh for apps you don't want using the processor. If that doesn't solve the problem, stop using the phone. It sounds dangerous to me. Allowing a Li-Ion battery to get hot creates a risk of a fire, and it also shortens the battery's life.