You can't really "upgrade" your graphics card itself. You can, of course, upgrade by buying a NEW graphics card.
The only way to increase performance is make sure your PC is running top notch. Meaning you have few startups, no programs running, no viruses and spyware, a clean system.
Some system optimizations would help, like OS performance tweaks and registry tweaks and swap file optimization. Keep yourself defragged as well.
Once you have a clean system, make sure you have all the latest drivers for your chipset and video card. Possibly BIOS and sound and network as well.
Once your drivers are up to date, make sure your games are as well, any patches that are out, etc...
Next, since you want to run newer games, you will simply have to turn the graphic settings down to lower performace. Run at a lower resolution and refresh rate in-game.
Then besides your graphics card, you want to have a good amount of RAM and CPU speed. About a gig of RAM is average for a gamer these days, just average. More is becoming popular anymore.
------------------
That being said, as for buying a whole new system, we have to start with budget. Because you can get a new computer from $300 to $10,000.
If you have no budget, I'd say go check out Falcon, and Alienware PCs. They make some of the best top-end gaming PCs. For a premium price of $4,000 to $8,000 and thereabouts depending on features.
Or for a tighter budget, get one built from someone like ABS Computers (
www.abspc.com, configured from about $1000 to $2000+.
If you are on an even tighter budget, you can forgo certain parts if you already have them, and build it yourself with just the necessary parts, for under 1K
But never would I recommened the likes of Dell, Compaq, HP, Sony, Gateway, E-machines, Fry's special, GQ, Walmart special, or any other name-brand cheapo. Unless it is just for grandma to get E-mail on, or a school kid to do homework.
There isn't really such thing as "inexpensive gaming PC". Gaming is what MAKES a PC expensive, because it requires more power then most any other function.
Anyway, enough from me. Hope it helps.