$600 is cutting it very, very close. $700 is more reasonable, especially for a monitor and all the peripherals.
Case:
NZXT BETA EVO Classic Series CS-NT-BETA-EVO - I got this since it is a simple mid tower with free shipping. $59 (Before $20 mail-in rebate)
Motherboard:
ASUS M4A785TD-V EVO AM3 AMD 785G HDMI ATX AMD Motherboard - A solid board that uses AM3 CPU's, DDR3 RAM, and Crossfire. I chose this because it has on-board graphics which means you do not need to purchase a dedicated GPU to run the PC. $91
CPU:
AMD Athlon II X2 250 Regor 3.0GHz - If there wasn't a tight budget, I would suggest one of the Phenom II's, but the Athlon II will work well for your application. $67
Power Supply:
Thermaltake TR2 W0070RUC 430W - Since you're not running a powerful GPU on this system, a simple power supply is all that is needed. This can easily handle the power requirements, but is also powerful enough to handle a GPU should you choose to install one. $36 (Before $12 mail-in rebate)
RAM:
CORSAIR XMS3 4GB (2 x 2GB) 240-Pin DDR3 SDRAM DDR3 1600 - More than enough RAM for most users. $92
Hard Drive:
Western Digital Caviar Green WD5000AADS 500GB 32MB Cache SATA - Plenty of space. $54
Keyboard:
Microsoft Comfort Curve Black 105 Normal Keys 9 Function Keys USB Wired Ergonomic Keyboard 2000 - A simple, basic keyboard. $13
Monitor:
ASUS VW198T Black 19" - Asus makes some very good monitors. This one also comes with speakers built in, though, audiophiles won't use them. $119
Mouse:
Microsoft Laser Mouse 6000 - v1.0 - I usually use either Microsoft or Logitech mice and this is a good one. $29
Optical Drive:
LG 22X DVD Burner - Bulk Black SATA Model GH22LS50 LightScribe Support - You need a basic drive. This one can read and write to most optical media. $28
Operating System:
Windows 7 Home Premium 64-bit - You need a 64-bit OS to use 4GB or more of RAM. $104
Grand Total: $701 (Before Rebates and Shipping)