Your computer may have 5.1 through the HDMI, but that is digital and unamplified. The only way you could get away without using a receiver is you
A) Buy a digital 5.1 PC Speaker Setup
B) Have analog connectors on your computer as well and use those with some adapters and regular speakers.
The built in amp for analog is probably crap, so if I were you I would buy a nice receiver and speakers and run the computer through that. Ideally get one with HDMI in and out, then you only run one cable to it, it decodes the digital stuff and makes it sound good and sends the video on to the computer. And you could add more devices later on and just use the receiver to switch them (game consoles for instance).