iPhone 5S overheating with very short battery life
The battery in your phone is dying. Continuing to use it and charge it could cause it to catch on fire. You can take it to Apple to be repaired/replaced but will probably cost a lot as it is an older phone or you can take it to a local phone "fix-it" place to have the battery replaced.

It looks like https://www.batteriesplus.com will replace it fot $60 or you could go the ifixit route and do it yourself.

I do recommend that you replace the battery soon, I had a 3GS years ago with the exact same symptoms, the battery eventually expanded so much it broke the back of the phone and could not be repaired.