Is your HP power supply DC or AC? If it is already DC, then you don't need the rectifying diodes (D1-D4) and can connect straight to the first capacitor instead. Just make sure to connect + and - the correct way around.
As to creating the negative, it can be done but not quite as simply as the positive regulator side. You would need to use an inverting, switching regulator. About the simplest (and a good one) is the MC34167 from OnSemi, although there are many others. See page 12 in the datasheet (second link) for the circuit. this is
just for the negative side - you would build the positive regulator as above, then connect your +12V to the Vin(12V) in the datasheet to get the negative output.
https://www.onsemi.com/PowerSolutions....do?id=MC34167
https://www.onsemi.com/pub/Collateral/MC34167-D.PDF
It would be easier if you could get another PSU or transformer, and wire the negative of the first to the positive of the second. That way you will get the +15, 0, -15 to feed to the regulator circuit in my first post. Just make sure that the PSU's are the type that are
not connected to the mains ground/Earth, i.e. they only have two mains wires, not three, otherwise you can't wire them up like that, something will melt! If you can get one, ask if you want more advice on this.
A couple of extra tips:
The 78xx series of regulators need at least about 2V more input than output, so 15V is a good choice to give a 12V output but not much less.
You might be surprised how hot the regulators get if you draw even a few hundred mA from them, and if you get up to an amp or so, really hot! You will need to use a heatsink. If your regulators/heatsink get so hot that you really don't want to keep your fingers on it, you need a bigger heatsink.