This sounds like a faulty inverter cable, which is fairly common with MacBooks. Relatively inexpensive, but somewhat difficult fix.
There is a guide already on iFixit - Inverter Cable Replacement
The replacement cable is about $20 through iFixit - Replacement Inverter Cable
A note with the cable... I would remove the MacBook's top case & look at the end of the inverter cable that connects to the logic board before ordering, just to ensure you order the proper cable. There are two options and I don't entirely trust how iFixit is differentiating the two. I've seen hundreds of the Santa Rose / Penryn model connections but only a handful of the Core Duo / Core 2 Duo connections.
Hope that helps!