Blank or black screens generally indicate a failure feeding data to the Graphics Processing Unit (GPU) in your video card. This can happen for a bunch of different reasons as different processes run as your system boots up. You may notice flickers during boot - these are indications that different processes are starting up and interacting with the GPU.
When the grub boot menu comes up, press e to edit the correct boot line.
Change
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash"
to
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash nomodeset"
If this works, then you can install proprietary graphics drivers which should get you going again.
If you have trouble with the above, you can change the above line permanently in the grub configuration file.
Boot into recovery mode (selection from the grub boot loader) and edit the file /etc/default/grub as per the above instructions.
You can edit this file by typing:
vi /etc/default/grub
You'll have to update the grub menu:
update-grub
then reboot by typing
reboot
You can also try acpi = off and nolapic if nomodset also shows up as a black screen.
Replace quiet splash with no splash to get an idea of what step your system is failing at.