What you're talking about isn't really "upgrading" your current PC, it's taking your current hard drive, case, and PSU over to a new build.
You can in theory just leave windows as it is on the hard drive and plug it into your new build and boot up, however, I would definitely recommend doing a fresh install of windows once you have the new PC built. This is primarily because of driver software. Right now, your copy of windows has lots of Packard Bell driver software installed that are designed to work with that PC's hardware. However, if you try to run this OS/drivers on the new PC, it will not be running smoothly as all the hardware is different. You will want to re-install windows and install the correct drivers from the DVD that comes with the motherboard, and then graphics drivers either from the DVD that comes with the graphics card, or from the Nvidia website. (Usually the website drivers are more up-to-date than whatever is on the DVD.
Should you use your current PSU for the new build? Probably not. Most OEM PSUs are built to be able to supply the power needs of the hardware inside them, and not much more than that. They are usually pretty bad quality as well. I would pick up a good quality 600W power supply for the new hardware, you don't want to over-load your 450W PSU and damage your new components!
Finally, the case. Again, I wouldn't recommend trying to install all the new hardware in the HP case. While it would probably work, the case is likely cramped and doesn't provide good cooling for your parts. The layout is probably non-standard and will be frustrating to work with.
Sorry, but I think the hard drive is the only thing worth carrying over from your current PC. That being said, you will find that a small SSD for your OS and programs will give you a MUCH faster desktop experience than a conventional hard drive. Most people will have a small SSD as the C drive and then a bigger hard drive for storage of files, games, etc.