The proper way to do the installation is through PPA. So copy/paste these lines in the terminal and it is installed.
This installation is a bit more specific because of line 2.
sudo apt-add-repository "deb http://repository.spotify.com stable non-free"
sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys D2C19886
sudo apt-get update
sudo apt-get install spotify-client