Cheapest 4 core i5 I can find is $180 and is the I5 2320,
http://pcpartpicker.com/part/intel-cpu-bx80623i52320, however it's over budget, and the only other ones that would keep up in my opinion would be the I3 3225, I would say step up to the I5 3245 but it's not that much better and it's $10 more. The 3225 has a base clock of 3.3 and the 3245 is only .1 more. Here is the link to the I5 3225
http://pcpartpicker.com/part/intel-cpu-bx80637i33225. I think it should work just fine, speed wise, for your gaming needs and budget. Good luck hope this helps