Well, if the computer is turning on, and just the screen isn't, it could be a few things.
Try turning it completely off and connecting an external monitor. Then turn on the external monitor and finally turn on the laptop.
Can you see on the external monitor?
If you can, then the attached display, or the ribbon cable that connects it are the problem. If you can't, then it is probably the graphics card/GPU that is the problem (and possibly the motherboard as well).
However, before doing any work either yourself or taking it to someone, make sure it is no longer under warranty. If it still is, and you do any work on it, that would void the warranty. In that case it would be better to contact the manufacturer.