Is it just the screen that won't come on, or the whole device? If it is just the screen. Try turning off the laptop, plugging in an external monitor, turn on the external monitor and then turn on the laptop. Do you see your screen on the external monitor?
If yes, then the problem lies in the attached display, and can be a few things. Then you would need to check the ribbon cable that connects the display. If that isn't the problem I would then shine a light at the screen, when the computer is on, and make sure it isn't the backlight (if it is, and you shine at the right angle, you will see a faint image of the screen that you should be seeing). If not these, then you probably need to replace the display.
If you see nothing on the external monitor as well, then your graphics card is probably shot.