If the PC fan really does have has a brushed motor, then you can just control it with PWM. However, all the PC fans I have come across use brushless motors, which don't work very well if you PWM the supply voltage.
You can buy PWM-controllable PC fans. They only cost a little more than regular PC fans. The specification says you should PWM them at 25KHz, which can be done using an Arduino, although most will probably work over a wider PWM frequency range than the spec says.