I'd say take it into a Apple Authorized Service Provider and have them run AST on it. It should, if the serial number hasn't been set, prompt them to serialize the unit. Otherwise, you're likely looking at having to get your hands on the serializer tool. You'd want to locate Blank Board Serializer v 1.1.1 .
Likely that either the display has died, or the video card. I would suggest hooking the machine up to an external monitor to see if you get video. If you do, likely it's the display. If you don't, then that would point to the video card. I've been replacing more and more of the displays as of late for customers, so my gut says it's probably that.