You probably won't see much benefit from having a dual-core server. Most of the modern Xeons have more than 4 cores, and even if you find a dual-core Xeon, a Core i3 or i5 would probably be about as fast for a lot cheaper. Considering your budget, I'd probably look for a PC with a Core 2 Duo. Since they've been around a while, there are older and newer models. The ones with older models are probably pretty cheap nowadays. Also look for PCs with "Pentium Dual-Cores." Make sure it's not Pentium D. Pentium Dual-Cores are a lot faster than Pentium Ds. I would try to aim for the Core 2 Duos though, since a lot of times, if a manufacturer uses a Pentium or Celeron rather than a Core 2 Duo, they're trying to cheap out, and the PC will have a bad hard drive or soon-to-be faulty RAM.
Be cautious when looking for OEM computers with AMDs. A cheap AMD CPU in an OEM computer is often another sign of the manufacturer cheaping out. I think you'll probably want to look for Athlon 64 X2s and Phenom II X2s. In the Athlon 64/Pentium 4 era, the Athlon 64s were actually faster than the Pentium 4s, so a lot manufacturers used them as performance options, not cheap out options. The Phenom IIs are AMD's current performance processors, so a manufacturer will probably not use them as cheap out options either. I would steer away from Athlon II X2s, since those fall more into AMD's current budget category.
