I'm not totally sure what you want; Do you want to build a PC yourself, or do you want to buy a whole pre configured one like those from Dell.
I can't really think of building a PC for under $350 that is even decent. You'd be better off buying a Dell, or an MDG.
You'd be kinda off like this
CPU: <>50 (I'd get an Intel Dual Core Celeron or Pentium Dual Core and overclock) [I'd go with a 775, cause you can go quads when you think this is getting old.
Motherboard: <>70 (Depends)
Ram: $30 (2x1GB DDR2-667) Higher the better for overclocking (I got OCZ PlatXTC 800 for $20)
HD: $50 (depends)
Case (w/ PSU): $50 (depends)
CD/DVD drive: $30 (Depends)
Video Card: $30 (6200, 7300, 8300 or other lower end, mainstream[6200, 8300 or 7300, you can attempt to flash a mid end x600 series BIOS (6200 -> 6600, 7300 ->7600 8300 ->8600 (OWN RISK, i never tried before)) For a little more, you may get a 8600GTXXX which can run CS: Source over 150 FPS (regardless CS 1.6). I have a 7600GT that runs over 160 FPS with some hogging anti virus program.
-> $310; Doesn't include monitor, nor keyboard, mouse, or taxes.