Let me get this straight. You need all of the following:
Not happening on $350. The OS alone is $100. The cheapest CPU is about $50, as is the board. RAM doesn't get cheaper than $50. A good case and PSU is at least $100. Right there is your entire budget. No GPU, no optical, no HDD, and definitely no gaming.
You really need at least $600 to start a decnet quality build. Now, if you don't need some of those things (mainly the OS and either GPU or CPU/mobo/RAM), then we can do something.
That's also a DDR2 version of the HD4650, which will be muttly. If a HD4650 is all the GPU you can afford right now, at least get a DDR3 version. Better still, stretch your budget to a HD4670.
Particularly assuming a future GPU upgrade, pick up a 430W Antec Earthwatts or 400W Corsair PSU at the least, although for 1920x1080 gaming, you will really want a GPU more likely to need a quality 500W PSU. Look for full range active PFC (no little voltage switch) and 80+ certification. Antec, Corsair, PC Power & Cooling, Seasonic, and Enermax are among the better brands.