Once the screen goes dark, push the home button and shine a bright flashlight on the screen. If you see a dim image, then the problem is backlight related. If you see no image at all, then you have an LCD issue. It's hard to say if its related to the actual LCD itself (which also includes the backlight) or with the driver circuits on the logic board.
Has anything out of the ordinary happened to the iPad recently, such as drops, damage, repairs or contact with water?
At some point, it will have to be [guide|36020|opened] up for proper troubleshooting.