Here are the steps to install and make Spotify client for Linux work on Ubuntu 15.04:
First, download the client following the instructions here: https://www.spotify.com/download/previews/
Then, follow the instructions from here: http://www.webupd8.org/2015/04/fix-missing-libgcrypt11-causing-spotify.html
If you then want to get your mp3 files play on the client you need to do this:
sudo apt-get install ffmpeg ubuntu-restricted-extras
After that, install the libs that you need by following this post: https://community.spotify.com/t5/Help-Desktop-Linux-Mac-and/Linux-Local-files-do-not-play-on-Ubuntu-14-04/m-p/1072805/highlight/true#M119043
Now things should properly work.