Won't turn on and charge but heats up

Its battery might be exhausted.

I suggest that you make an appointment (*) and have your device inspected at an Apple Store or AASP (Apple Authorized Service Provider) among this list:--> Apple – Location Selector. After checking it out, they'll give you all the available options.

Regards

Giulio

(*) You can do it:

• Online:--> Genius Bar Reservation and Apple Support Options - Apple;
• Contacting a local Apple Store or AASP among the list provided above;
• Using the "Apple Store" app, available and downloadable from AppStore.

iPhone 5c is an outdated iPhone (> 8 years), which also affects the battery life.

What happens when you connect the iPhone to a PC?