The -12V lead from a PC power supply only goes to the motherboard, so you either have to splice into this lead, or be satisfied with the +5V and +12V you can get from the drive connectors. That either limits your chip choice and output voltage severely, or you require a DC-DC converter to bring the voltage up.
Through suitable filtering, you can clean up the noise from the power supply. You wouldn't want to plug it straight into an op-amp, but it wouldn't require heroic measures to get clean enough power to get reasonable sound quality. You'd be more worried about RFI pickup than power supply quality, I'd think.
If you add a high-quality DC-DC converter, you effectively get filtering "for free" -- the converter's job is to put out a stable, clean voltage as long as the inputs are within range. A good DC-DC converter will cost you about $20, though.