Sounds like either the battery or board is shorted.
At this point first try holding home + power button for 10 seconds and try turning it on again.
If it still does not work then I would try another battery and if that still does not work then you might want to take it to someone that does micro soldering as it sounds like a board level issue.
Usually iPhones don't suddenly stop responding and get hot for no apparent reason so I would say the screen assembly and anything else connected to the board via ribbon cables aren't at fault other than maybe the battery.
Guide to replace battery:
How to replace the battery in your iPhone 5s
Where to buy an replacement battery:
iPhone 5s Replacement Battery