How to build a decent gaming PC out of an old computer?

There's nothing to be done with what you have. You can't add a video card because it won't fit and your power supply is tiny and no after market power supply large enough to power a graphics card will fit in your case.

You would first have to get a new case, if your board is not custom to fit only that case, then a power supply then a graphics card.

This might work - http://www.newegg.com/...6814487027
http://www.newegg.com/...6811147010

East, Scrao it and buy a deceht one.

First of all, since a 60Hz monitor can only display 60fps, 80fps on the screen is still only 60. Second, online has internet communication factors. Third, most people can't see the difference over about 40fps.
Your PC: http://support.hp.com/.../c01466473
Now, your current PC is about the worst possible. Slimline means custom power supply and only low profile graphics cards. Your power supply is old now, and only 160 watts.
The CPU in there's 2005 to 2006, and no doubt the system is close to that old including the power supply.
Given operating system Vista, and the specs, it may pay to buy a used system as either tower or laptop.
Your system was supposed to have 2x2GB RAM as sold as the maximum allowed in it.
CPU lists as Athlon 64 X2 5400+, but given the ram issue, you also may have an even lower end CPU.
This one 5400+ at 1432 score is in line with new cheapest laptops
https://www.cpubenchmark.net/...u_list.php
Your graphics score is the weakest part of the system. The motherboard graphics you have scores only 22
http://www.videocardbenchmark.net/gpu_list.php
A $110 GTX 750 in a normal entry level gaming PC scores 3246, which is almost 150 times the graphics power.
Belarc Adviser and Speccy tell people what they have.
Belarc does work on Vista
http://www.belarc.com/...nload.html
Speccy may also
https://www.piriform.com/...com/speccy
Must know your processor and RAM.
If you added a low profile GT 720 and have the 5400+ and 2GB RAM. You should get smooth play on high settings and don't worry about fps if it reads 40+
That would be the only upgrade and if it doesn't fix it, you sell it all and need a new PC.
http://www.game-debate.com/...quirements
GT 720 is 19 watts, and with that old dismal supply, I wouldn't try much more than that.
http://pcpartpicker.com/parts/video-card/#c=183&i=39&sort=a8&page=1
In the USA, under $58 new, and that is more than your entire PC is worth today without the card.
Can't say what it does online, because of internet interfacing. It calculates to over 60fps on paper.
Maybe find a used card on ebay. It must be low profile. You need low wattage.

Keep the power cord. Replace everything else.