This is actually pretty simple. Anybody can do this quickly.
Go the the HP Driver web site. Click the link below
Type in your printers name or model (eg. HP Officejet 4620)
These instructions are good for any HP printer that will not work with Windows 8
A couple of choices might come up for which printer to choose. Pick the correct one.
The choice you want to pick here has a disk next to it and says Software & Driver and Downloads
Put in Windows 8 as your Operating System. To find out if you have 32 or 64 bit click on Start (bottom left of screen) then go to Control Panel\System and Security\System then its under system type
The driver you need is under HP ePrint Software