(1) Check the battery by replacing it (2) Could also be the front facing camera cabled could be bad or not properly connected to the board, I have seen the device get hot from a bad front camera, but that doesn’t explain why it wont power on, I would replace the battery first.
Sound like a software problem or you maybe low on system storage, does it ever tell you your start up disk is full? I would try to backup your data and then do a wipe and reinstall of Mac OS X