360 watt psu and Q6700 QUAD with a 9600GT would be pushing the limits. I believe you should get a new power supply for the HP. I don't think that they use a proprietary power system like dell, so everything should fit in normal if you got a new, more powerful one. If you don't have much cash on hand, just get like a 400-500watt Cheap PSU from a Known good manufacturer such as PCP&C. Don't get some $10 unstable reactor ready to meltdown.
You could get a StealthXStream 600W PSU if you want a really good video card and your PSU holds you back. I'm suggesting it because it's pretty good quality, cheap, and I actually fit one in an HP once. PSUs like 520HX or Silencer 610W wouldn't have worked in that smallish case. Maybe yours is different, I don't know. What size is your current PSU (length, width, height)?
How much clearance do you have now, with the 8600GT? That is, if the card was 4 inches longer, would it still fit? The best video cards available these days are almost 11" long, and IIRC the 8600GT is under 7".