Did you try booting into the recovery mode (selected via grub)? You could then try to restart the upgrade with
sudo apt-get update
sudo apt-get upgrade
You may need to run
sudo dpkg --configure -a
to fix the partially installed packages.
If the recovery mode is also broken, you could try booting off a live cd and then chroot'ing into your hard drive installation and performing the above steps.
To chroot into your hard drive installation, first boot off a live cd. You can use the ubuntu install CD for this task, just choose the option "try ubuntu" rather than installing. Once booted, open a terminal and mount your root partition. If your root partition is /dev/sda1:
sudo mkdir -p /media/ubuntu
sudo mount /dev/sda1 /media/ubuntu
Then you need to mount the proc, dev, and sys filesystems into your new environment:
sudo mount --bind /dev /media/ubuntu/dev
sudo mount --bind /proc /media/ubuntu/proc
sudo mount --bind /sys /media/ubuntu/sys
cp /etc/resolv.conf /media/ubuntu/etc/resolv.conf
Finally, chroot into it:
sudo chroot /media/ubuntu
From here you can run apt-get and dpkg to fix your system. This is identical to the way that people fix grub using chroot, in case you run into problems you can google something like "grub chroot ubuntu livecd" and find more info.