Have you confirmed that it's your back light not turning on is the problem? If it is, you ought to be able to just barely make out some details on the screen if you shine a bright light on it when you're logged in.
So assuming the back light is the problem, here's what i did to fix the problem on my notebook. Enter this command in the terminal:
When your grub file comes up, paste these two lines at the end of the file, where you see similar code:
gksudo gedit /etc/default/grub
then enter this command:
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash acpi_backlight=vendor"
This will save the changes.
Now, when you reboot, the brightness keys on your keyboard ought to work. On my notebook, the keys are "Fn" and "arrow down". Assuming the fix works, you'll have to adjust the keys every time you reboot.
HTH, and good luck.