So will this build be able to play most games?
Not really. I think with your budget, you will have to settle for an AMD build.
It will be ok for gaming, but I just thought I would share my experience of switching from AMD to Intel, as there seems to have been quite a lot of talk on the forum recently that gaming performance between the two is roughly the same.
What was your GPU setup? If it was your 2 580s, then yeah, an i5 or i7 will get better framerates compared to AMD. But when you have a less powerful GPU, games rely less on the CPU and more on the GPU. My setup gets 60+ fps on BF3 on ultra, with my AMD CPU.