In the stickies up the top of this forum, Xzin has posted his Wiki which includes some detail on what types of minimum hardware would be recommended.
Multiboxing Wiki - My 5 (Now 10!) Boxing WoW Writeup

I've been curious about this myself, tempted to try it on my rig: E8400, 4Gb ram, 2 x 8800GT's. From the brief amount of reading I've done here it looks like the more instances of WoW you run, the more you're memory is severely taxed. Given that RAM is so cheap I'm tempted to go 8Gb

Obviously if it's a serious attempt then most would recommend dedicated hardware. Also there MAY be the possibility of OC'ing to see if you can squeeze more performance out of your rig, but that would mean upgrading your cooling system at the least.

Post the specs of your rig as a start anyway to see if its remotely possible!