Three terminal-commands:

sudo apt-get install build-essential cdbs fakeroot dh-make debhelper debconf libstdc++6
sudo apt-get install dkms libqtgui4 wget execstack libelfg0 dh-modaliases
sudo apt-get install linux-headers-generic xserver-xorg-core libgcc1

Additional two terminal-commands for 64-bit:

sudo apt-get install ia32-libs lib32gcc1 libc6-i386
cd /usr ; sudo ln -svT lib /usr/lib64

(Note: The second command shouldn't be necessary if there is already such a symbolic link named lib64pointing to folder lib there.
And if there is already a real folder by that name you should ensure that its contents are safely moved into folder /usr/lib and then delete --now
empty-- folder /usr/lib64 before executing this command.)

Download from these direct-links:

http://www.mirrorservice.org/sites/archive.ubuntu.com/ubuntu//pool/main/u/udev/libudev0_175-0ubuntu13_amd64.deb

and (the older intel driver - xserver-xorg - see also section "Important" on answer for 12.10. I think this would also be very important here.)

https://docs.google.com/file/d/0B0tTaH4qTIIXdDF3NThFWUtrYlU/edit?usp=sharing

Copy these files (two .deb packages) into an empty folder.

Remove the old drivers:

sudo apt-get remove fglrx*
sudo apt-get remove xserver-xorg-video-intel

Execute the following two terminal-commands in the folder with downloaded .deb files:

sudo dpkg -i libudev*
sudo dpkg -i xserver-xorg-video-intel_2.21.6-0ubuntu4_amd64.deb

Installation:

Get the current ATI Catalyst driver e.g. 13.04 (I have tested it with this release) from the official website: www.amd.com/drivers

Unzip the .zip and make it executable. Then go to the folder with the unzipped .run-file in terminal and type:

sudo sh ./amd-catalyst-XX.X-linux-x86.x86_64.run --buildpkg Ubuntu/raring

Replace XXX with the correct name of the file.

Install the created .deb-files with the following terminal-command in the current directory:

sudo dpkg -i fglrx*.deb

Post-Install:

Enter the terminal command

sudo mv /etc/X11/xorg.conf /etc/X11/xorg.conf.backup
sudo aticonfig --initial -f

(Note: If you use multiple AMD graphics cards or AMD dual graphics, you can use sudo
aticonfig --adapter=all --initial -f instead, and/or if you use a dual monitor display, you can also use this additional third command: sudo
aticonfig --set-pcs-str="DDX,EnableRandR12,FALSE")

Select the discrete GPU

sudo aticonfig --px-dgpu

Reboot your system

sudo reboot

Install the Unity:

dconf reset -f /org/compiz/
gtk-window-decorator --replace