Another attempt would be trying to upgrade/fix your PC via the Recovery mode's Root Shell (if it allows you to boot that successfully).
From Grub menu, choose Recovery Mode (the option below the normally used one probably) and then choose Drop to Root Shell. Run these commands one by one:
Where 'eth0' is your network interface. This command would connect your PC to the internet.
If the commands finish without any errors, hopefully your install would be repaired and you might be able to boot it successfully.
sudo apt-get install -f
sudo dpkg --configure -a
sudo apt-get update && sudo apt-get upgrade
sudo apt-get dist-upgrade
If the commands don't work, the only choice is to boot an Ubuntu Live CD/USB and then mount your partition and copy over your data and do a fresh install.
If you are unable to even get to the Root Shell, you can try to Chroot your install from a Live CD and then run the same commands as listed above.
To Chroot, boot an Ubuntu Live CD/USB and get to a Terminal, then run these commands one by one:
Replace 'XY' with your intended partition to which Ubuntu is actually installed. It should actually look something like 'sda1', 'sda2' etc. You can find it by running this command:
sudo mkdir /mnt/temp
sudo mount /dev/sdXY /mnt/temp
for i in /dev /dev/pts /proc /sys; do sudo mount -B $i /mnt/temp$i; done
sudo cp /etc/resolv.conf /mnt/temp/etc/resolv.conf
sudo chroot /mnt/temp
Once Chrooted, run the same commands as mentioned above.