Normally I would suggest replacing the battery as the first step. How was the battery life before this occurred? Was it good and then the phone just died? Or has the phone been gradually getting less and less time on each full charge?
Try to connect your phone to an iTunes enabled computer. Does iTunes recognize your device? If itunes does recognize your device, then shine a bright flashlight on the screen. Do you see a dim image? If so, you have a backlight problem. If you don't see an image, then you probably have an LCD issue. Both of these problems can be caused by either a bad screen or logic board issue. At that point, you will need to try a screen to see if the problem is solved or not.
If iTunes does not recognize your phone, then it could be a bad battery or a logic board issue. Unfortunately, the only way to troubleshoot this is to try a new battery. All of this requires that you buy parts. It might be simpler to have it looked at by a local repair shop.