I tried this years ago myself, and after scratching my head trying to figure out why the LM7905 wasn't producing -5V from my +12V supply, I finally realised it doesn't create negative voltage, it can only reduce an already existing negative voltage. So you need to run it off -12V in order to produce -5V.
And of course as SquallStrife says, the output you want is -5V with respect to the chassis as GND, which means the input has to be -12V or so with respect to the chassis as well, otherwise the two GND voltages will differ. You'll get -5V out of the regulator with respect to the input GND (which is +12V) but you'll get something completely different once you try to use that with respect to the chassis GND. I'm not actually sure what voltage you would get as I don't know how to calculate it. @SquallStrife any ideas?
If you disagree, go ahead and build a circuit and you will see - the parts are cheap enough. If it works we will all buy one from you 😀
The thing is to remember that voltages don't exist in isolation. You can't have a voltage on one wire. Voltage is a measure of the difference between two different wires (that's why you need to connect both multimeter probes to measure a voltage. If you connect only one probe then there is no reading.) For PC power supplies, the second probe is always connected to chassis GND, and you make your measurements with respect to that point. The proposed circuit only produces -5V when you have moved the GND probe somewhere else in the circuit, which nothing else in the PC uses as GND, which is why the reading will change when you go back to measuring GND from the chassis again.
It's a bit like sitting on a chair in the bottom of a deep hole and saying you have the shortest chair in the world, so short that the seat is at a negative altitude. But once you bring that chair back to the dinner table, suddenly it's no shorter than the others. You only get an accurate measurement when the ground the chair is sitting on (GND) is the same for all the things you are measuring.