Unfortunately, the specs page for your motherboard does not provide a full CPU support list. While the best processor for Socket 775 is the Intel Q9650, whether or not it actually will work is a guess.
For full HD videos and games, you're looking at discrete graphics card (and possibly a new power supply).
Given the cost of these two (or three) upgrades, I'd probably recommend starting from scratch and building new.
Edit: based on this web site, the best processor the G31 chipset can take is the Intel Core2Quad Q6700.