It sounds like the charge port on the phone took a dump to be honest.
Samsung charge ports seem to die easy.
Mine died and that is what it did. The phone would vibrate every 10-20 second when it was plugged in :/ if I tapped the power button it would show a gray battery symbol, not the green which it should if the phone is off and charging.
If you plug it into a pc, does the PC beep to recognize it?>
The nice thing is if your out of warranty, google "replacing charge port S4" It is REALLY easy I did mine in about 15 minutes. (MAKE sure you order the right port if you do this, the different S4 versions have different charge boards)