Yeah, it's pretty tough with that budget.
Jaraldo's recommendation to save a bit more is a good one. The problem with such a limited budget is that at that low end of the market, every extra $ goes a long, long way performance wise. It's also hard to build something that you can efficiently upgrade down the track.
If you really want to start now, honestly, I think you'd be better off getting a reasonable second hand computer and adding a video card. I'm always nervous recommending things of ebay, but something like this would be a decent start:
http://www.ebay.com/itm/HP-DC7600-Tower-Dual-Core-3-4GHz-4GB-Memory-250GB-Combo-Computer-/321069929160?pt=Desktop_PCs&hash=item4ac14252c8
You could then add a video card, I'd recommend the GTX 750 ti because it should play happily with any powersupply AND, for your case, is compatible with "Shadowplay" which enables recording while gaming and offloads the bulk of the work to the GPU hardware. Basically you get the recordings with a much smaller hit on your CPU while gaming.
This MSI one is a bargain at moment at %30 off after MIR:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814127784&cm_re=gtx_750_ti-_-14-127-784-_-Product
Just make sure whatever computer you get has at least 4GB RAM and supports full sized PCIe cards... go to the manufacturer website if required.
You may also need to budget for Windows 7 or 8 if the computers only have XP. That wouldn't be ideal!
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