Usually on older systems you put the new bios on a USB stick and boot from that stick. Then the bios updates, and you will reboot after removing the stick.
I'd just try the CPU without updating the bios. From looking around at the HP site, it sounds to me like support for that CPU was included from the start.
One bit of advice, get an SSD if you don't already have one. It will make your computer a lot more usable. A small 120gb for your OS and a few key programs is all you need. Sandisk makes some cheap ones that work alright, like the 'Plus' and 'Ultra' series.