For 13.04:

Pre-Install:

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 lib64 pointing to folder lib there. And if there is already a real folder by that name (determined with ls -l /usr/lib64), you should ensure that its contents are safely moved into folder /usr/lib and then delete --the now empty-- folder /usr/lib64 before executing this command).

Download from these direct-links: 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

Prevent xserver-xorg-video-intel from being upgraded:

sudo apt-mark hold xserver-xorg-video-intel

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

Links:

http://ubuntuforums.org/showthread.php?t=1930450

My Scripts for Switching

Script for High-GPU-Mode:

#!/bin/bash
# Activate discrete GPU (High-Performance mode), must re-start X to take effect
sudo aticonfig --px-dgpu
sudo restart lightdm

Script for Low-GPU-Mode:

#!/bin/bash
# Activate integrated GPU (Power-Saving mode), must re-start X to take effec
sudo aticonfig --px-igpu
sudo restart lightdm

Script for showing current GPU-Mode:

#!/bin/bash
#Show current GPU (High- or Low-Performance mode) as notification
aticonfig --pxl | while read SPAM_OUT; do notify-send "$SPAM_OUT"; done

Note: If you always have problems after the execution of a Switching-Script, make a restart of your system (and not only a re-login).

New!!! INDICATOR for Switching:

To switch between Discrete (AMD) and Integrated (Intel) GPUs nice and easy way, go to https://github.com/beidl/amd-indicator, and then download and install the DEB file there.

For 12.10:

It works for me!! And here is the how-to which I created for you:

Pre-Install:

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

Optional if 64 Bit - two terminal-commands:

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

Download from this direct-link: https://launchpad.net/~andrikos/+archive/ppa/+sourcepub/2755647/+listing-archive-extra the files and this two .deb packages into an empty folder

xserver-xorg-video-intel-dbg_2.20.0-0~andrik1_XXX.deb
xserver-xorg-video-intel_2.20.0-0~andrik1_XXX.deb

where XXX should be your architecture identifier (x86 or amd64)

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

sudo dpkg -i xserver-xorg-video-intel*.deb
sudo dpkg-reconfigure Xorg

Then reboot your machine

Note - this is from the PPA: https://launchpad.net/~andrikos/+archive/ppa/+packages?field.name_filter=&field.status_filter=published&field.series_filter=quantal

Important - Today I got a security-update for "xserver-org" form the official Ubuntu repositories which crashed my system again (no login screen). Then I installed the newest two xserver-org-video-intel*.deb's (downloaded and installed as described above) from the PPA https://launchpad.net/~andrikos/+archive/ppa/+packages?field.name_filter=&field.status_filter=published&field.series_filter=quantal again. You can also add this PPA on your system for preventing this issue. For me, this PPA has too much other packages, so I will do it (downloading the two .deb-files and installing them) manually. An other solution is, to de-select the "xserver-org"-packages if there are official Ubuntu security updates available.

Installation:

Get the current ATI Catalyst driver e,g 12.11 Beta (I have tested it with this release):

wget -c http://www2.ati.com/drivers/beta/amd-driver-installer-catalyst-12.11-beta-x86.x86_64.zip -O catalyst-12.11-beta-x86.x86_64.zip

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

sudo sh ./amd-driver-installer-XXX.run --buildpkg Ubuntu/quantal

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 aticonfig --initial -f

Reboot your system

sudo reboot

optional - fixing the bug for direct rendering on the integrated card:

gksu gedit /etc/X11/Xsession.d/10fglrx

Add the string "/usr/lib/x86_64-linux-gnu/dri/" on your 64Bit system that the line finally looks like this:

LIBGL_DRIVERS_PATH=${LIBGL_DRIVERS_PATH}:/usr/lib32/fglrx/dri:/usr/lib/x86_64-linux-gnu/dri

Add the string "/usr/lib32/dri/" on your 32Bit system that the line finally looks like this:

LIBGL_DRIVERS_PATH=${LIBGL_DRIVERS_PATH}:/usr/lib32/fglrx/dri:/usr/lib32/dri

Links:

http://ubuntuforums.org/showthread.php?t=1930450

http://ubuntuforums.org/showthread.php?t=1930450&page=51

http://www.upubuntu.com/2012/10/install-amd-catalyst-1211-beta-driver.html

My Scripts for Switching

Script for High-GPU-Mode:

#!/bin/bash
# Activate discrete GPU (High-Performance mode), must re-start X to take effect
sudo aticonfig --px-dgpu
sudo restart lightdm

Script for Low-GPU-Mode:

#!/bin/bash
# Activate integrated GPU (Power-Saving mode), must re-start X to take effec
sudo aticonfig --px-igpu
sudo restart lightdm

Script for showing current GPU-Mode:

#!/bin/bash
#Show current GPU (High- or Low-Performance mode) as notification
aticonfig --pxl | while read SPAM_OUT; do notify-send "$SPAM_OUT"; done

Note: If you have always problems after the execution of a Switching-Script, make a restart of your system (and not only a re-login).

For 13.10

it works without tweaking on a fresh Ubuntu 13.10 install:

1) Open the terminal and type

sudo apt-get install fglrx fglrx-pxpress

2) and reboot

For 13.10:

Pre-install:

Execute 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 xserver-xorg-video-all 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

If ia32-libs is not available, follow these directions and install with:

sudo apt-get install libgtk2.0-0:i386 libpangox-1.0-0:i386 libpangoxft-1.0-0:i386 libidn11:i386 libglu1-mesa:i386

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

Remove the old drivers:

sudo apt-get remove fglrx*

Reinstall the Intel driver:

sudo apt-get install --reinstall xserver-xorg-video-intel

Download the .deb package:

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

Execute the following terminal-command in the folder with downloaded .deb file:

sudo dpkg -i libudev*

Installation:

Get the current AMD Catalyst driver from the official website

Unzip the archive and make .run executable:

unzip amd-*catalyst-*.zip
chmod +x amd-*catalyst-*.run

Generate the driver packages:

sudo sh ./amd-*catalyst-*.run --buildpkg Ubuntu/saucy

Install the generated .deb packages:

sudo dpkg -i fglrx*.deb

Post-install:

Enter the following terminal-commands:

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 the following command instead:

sudo aticonfig --adapter=all --initial -f

Note: 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

Scripts for switching:

Script for High-GPU-Mode:

#!/bin/bash
# Activate discrete GPU (High-Performance mode), must re-start X to take effect
sudo aticonfig --px-dgpu
sudo restart lightdm

Script for Low-GPU-Mode:

#!/bin/bash
# Activate integrated GPU (Power-Saving mode), must re-start X to take effec
sudo aticonfig --px-igpu
sudo restart lightdm

Script for showing current GPU-Mode:

#!/bin/bash
#Show current GPU (High- or Low-Performance mode) as notification
aticonfig --pxl | while read SPAM_OUT; do notify-send "$SPAM_OUT"; done

Note: If you always have problems after the execution of a switching script, make a restart of your system (and not only a re-login).

New! Indicator for switching:

To switch between discrete (AMD) and integrated (Intel) GPUs nice and easy way, go to the project's GitHub page, and then download and install the .deb file from there.

Links:

1. [all variants] AMD/Intel Hybrid Graphics works — Ubuntu Forums
2. Ubuntu Saucy Installation Guide — cchtml.com
3. BinaryDriverHowto/AMD — Ubuntu Community Help Wiki
4. X/Troubleshooting/VideoDriverDetection — Ubuntu Wiki

This answer is based on Marian Lux's 13.04 guide.

I have a Vaio VPCSE with AMD/Intel hybrid graphics, too, and I followed your how-to step by step. Unfortunately, when I changed to integrated graphics in Catalyst, I got lost after login just as you desribed. When changing to discrete graphics, it works – but I can use fglrx from the sources for that.

So my only workaround, which I'm not quite happy with, is to use the open source drivers and switch off power of the discrete card using switcheroo. That is, with fglrx cleaned off my system, I type

sudo -i
echo OFF > /sys/kernel/debug/vgaswitcheroo/switch

assuming that

cat /sys/kernel/debug/vgaswitcheroo/switch

shows a + in the line containing "IGD"