First you should add the Spotify repository to a new file. Open the file with vim
sudo vi /etc/apt/sources.list.d/spotify.list
And add the following line:
deb http://repository.spotify.com stable non-free
Now you need to verify the packages with the key server:
sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 4E9CFF4E
And install the app:
sudo apt-get update
sudo apt-get install spotify-client-qt
Now you need to add three libraries to make playback work. The libraries are somehow missing from the Ubuntu 11.10 libraries, so you can try to find them somewhere else.