1.) If you use Raspbian - that won't work, since it is build using the hard float ABI capability of the ARM. If you try to install libspotify and make the examples work, eveything seems find at the beginning, but then you get an error like
libspotify.so.12.1.51: cannot open shared object file: No such file or directoryThis is due to the fact that libspotify seems to have been build using soft-float ABI. As long as spotify doesn't release a hard float build, you will have to go to step 2. This insight is the crucial part of the game here (and a "cannot open shared object file" error is not an obvious hint in this direction)
2.) If you want libspotify to work, you will have your RPi running the soft-float build of Raspbian, also available here: http://www.raspberrypi.org/downloads
3.) Once you have that, things are straight foward. Download libspotify from the spotify developer page and follow the instructions in the readme. This is also where you will need a spotify premium membership in order to get an appkey.
4.) In order to test it, you can use the jukebox example. Simply, after building it, run jukebox/jukebox. It will ask you for your login credentials and a playlist to run. If you don't hear anything, try another playlist. This "terminal version" of spotify seems to not tell you when a title is not available anymore, but instead simply keeps silent.
Advice: The jukebox example requires you to have alsa installed and *configured*. So, before testing the spotify api and complaining that it does not play any sound, you should configure the sound card. See e.g. here or simply google for raspberry pi and alsa Have fun!
PS: As a kitchen-radio this is still a bit uncomfortable. What I would ultimately like to have is a LAN-internal web-interface to the pi and libspotify, so that from every computer/tablet/smartphone in the LAN, I can access the local web-interface and search for/play titles, artists, albums ...