I'm using the EtherCard library and the "Backsoon" example just to get started with this module, but when I open the serial monitor to get the IP it just says "[Backsoon]" and doesn't show an IP
Most probably a wiring problem. As you failed to provide a wiring diagram we're not able to help you further. Also post a link to the exact hardware you're using (there are dozens of boards with this chip and my impression is you're not using the chip directly), better a link to it's schematics.