To address your points, hartmisc:
The short answer is that, driver wise, you're best bet is likely to ignore HP completely and visit the manufacturer websites for drivers for the individual pieces that need them. Several bits of hardware will be automatically installed by Windows 8 (I don't recall there being any especially exotic hardware in these laptops), but it's fairly easy to track down any that are "unknown" by using the Hardware IDs from Device Manager (or whatever the new Win8 name may be for it).
Addressing hardware requirements for Win8 in specifics, as Saugen did a bit of:
CPU - Depending on your processor, you may have some headaches. My C500 has a Core 2 Duo thanks to my extensive tinkering and modification. If you're running the stock Celeron M 420's that HP's website says for the C303NR, it's likely going to be uncomfortable. I ran Win7 a bit on my stock Celeron M 430 and it WORKED but it was not comfortable at all.
RAM - 2 GB is definitely needed. Depending on your motherboard (see my above post), that may be the most you can run without tinkering. It will be enough, but again may not be the most comfortable.
Video - Your video is an Intel GMA 950, and potentially the slowest of the lot (there are three levels in the C300/C500 line that I know of), so that will not be great either. My GMA 950 in my C500 handled Win8 just fine, but I also had more CPU muscle and more RAM, so your mileage may (and likely will) vary.
HD - Not much to say here except that if you're running stock it may get a bit cramped. A 7200 RPM drive would help but isn't 100% necessary.
So that's my input. If you're running everything STOCK, I would agree with Saugen in that your prospects are not great. It WILL run, but it may not be a good experience. If you're up for some tinkering, find out which motherboard you have and go from there. Parts for upgrades can be fairly cheap for these machines, provided your system will support them and you have the know how to install them.
If you need anything else, just pop back in. Someone will jump in!
O Shadow Fox O