Okay. 2.0 ghz dual core is (arguably) better than a 3.0 Ghz single-core. A single processor is easily locked up, and all of your processes are forced to run on the same core, so it isn't hard to fill the whole 2.6 Ghz with only a few tasks. Dual core is immensely better because it's much, much harder to lock up your computer. I had a 3.0 Ghz single-core, and stepped up to a 2.4 Ghz Quad core. TOTALLY different. Multiple cores changes everything.
Trust me, you want a dual core. And they're so cheap now, there's really no excuse. Especially AMD, though Intel's got some good budget options. (BTW, remember that you can't use an AMD processor in an Intel socket, and before you buy a processor, be sure it's the same socket as your motherboard. I think yours is a 478, which limits your options severely.

Long story short, if you really want to play games like Bioshock and want to spend as little as possible, get an XBox 360. It'll cost you less.

To play Bioshock, you'll NEED 2 GB of RAM, you'll NEED a relatively recent video card, and you'll want a dual-core processor.
The RAM is the easy part. $50 buys you 2GB of fast RAM.
The video card means you need a new PSU. Count on a minimum of $280-300 for this.
The processor would mean a new motherboard, one with a 775 or AM2 socket, and would run you $150-$400 depending on how fast you went.
Seriously, getting an entry-level system up to gaming level would cost you more than just buying a console and a few games.