If you are sure that you connected the external screen properly, it could be the video card or the motherboard. Try booting the laptop with the monitor attached. If you get no image on that, does anything come up on the laptop screen? Does it at least get to the BIOS?
If the drive seems to be spinning, and you can hear the Windows startup music, it may just be a screen issue, in which case replacing the inverter or the LCD panel will fix this.