My Inspiron 8600 has had a similar problem for at least six months. It boots just fine, then right after the Desktop has fully loaded, the LCD turns completely black. When I connected a remote monitor (CRT) to the port on the back of the laptop, I found that the Desktop displayed normally, and the computer was fully usable. Further experimentation showed that if I would press the Fn key, and then the F8 key (CRT/LCD), the laptop screen would be restored while the remote monitor would go black. However, in about 10-15 seconds, once again the LCD would turn black and the CRT would show the Desktop.
I've run antivirus scans with my AVG program, as well as four other online scanners, and all have found nothing more than three "tracking cookies." I've uninstalled/reinstalled my video card (nVidia GeForce4 4200 Go), as well as updated its drivers, all to no effect.
I've been using the laptop while it's connected to the remote monitor, but it's pretty hard on the back to carry the 21" Viewsonic CRT along with me when traveling.
If anyone here has found a solution that actually works, there are more than a few of us who would appreciate hearing from you.