Just a few suggestions. If you have the computer inside a computer desk, get it out and up off the floor. It will keep the dust out better and will also run cooler. The computer needs fresh air to keep everything cool. The CPU and GPU are cooled by ambient air temps, so give it as much as possible. If you have the option to put in another or better intake fan, try that. But, the computer needs to have fresh air and be able to get cooler air in and hot air out. That is true for any computer.

Whatever video card you choose, check the condition and size of your PSU. Your computer has a little age on it and PSU's tend to decline with age, kind of like me. You need a card that combined with the rest of your computer does not use 100% of the computer's Power requirements. I like to stay in thr 60-70% range. Your PSU will run cooler and last longer, plus you will have room for upgrades in the future. According to your specs, your computer has a 460W Power Supply. Typically, HP does not use high quality PSU's, so consider that. The best card, in my opinion, on the cards Britton30 gave you is the EVGA GTX 660 TI. In my opinion, price/performance it is one of the best buys on the market today. The problem is, it is considerably over your budget and rquires a 450W PSU. Which means, it would probably cost you a PSU too.

Failing that, the Sapphire 7750 is well within your price range, gets good reviews, but is calling for a 400W PSU. Depending on your usage and the condition of your PSU would probably work. But you will also have to keep in mind that a PSU is possible. But, considering the activities you have mentioned using the card, it will never be pushed to it's max power requirement. But, before making a decision, I would look at the PSU and try to determine the make (HP will probably have rebranded it to their name) the total wattage and the amps on the +12v rail. It should tell you that on a sticker on the side of the PSU. Most of the AMD 7000 series or the Nvidia 600 series are more power efficient than their previous generations cards.