My iPhone 4 with a new battery turned of by itself and wont turn back on

On Tuesday my iPhone 4 turned off by itself, I had tried holding the Home button and Lock button for 10 seconds and it turned on but my screen wouldn't work, the phone would not charge and it kept coming up on my screen that iPhone is to hot and I had to let it cool down even when it was not hot. So I got the battery replaced and I literally only got my phone back today and then at about half 6 the phone turned off by itself again. I have tried charging it, I tried holding the home and lock button and it still will not turn back on. Please some tell me they know whats wrong with my phone or what I can do to turn it back on.