If you need an OS, a $350 gaming machine is a really tough ask. It'd also be good to know what you want to achieve. GTA5 is a bit of an unknown, but your desired display resolution would give a good idea of what you might need to spend.
It's also worth considering that if you can afford to spend say $500, you stop looking at the "budget" market and are properly into decent gaming machines. As an illustration, here's a sub $500 machine that lacks in a few areas but packs in an i5 and a HD7850. PCPartPicker part list / Price breakdown by merchant / Benchmarks
If you need an OS, a $350 gaming machine is a really tough ask. It'd also be good to know what you want to achieve. GTA5 is a bit of an unknown, but your desired display resolution would give a good idea of what you might need to spend.
It's also worth considering that if you can afford to spend say $500, you stop looking at the "budget" market and are properly into decent gaming machines. As an illustration, here's a sub $500 machine that lacks in a few areas but packs in an i5 and a HD7850. PCPartPicker part list / Price breakdown by merchant / Benchmarks
Note that all of these builds heavily exploit deals/rebates to get the price down, so they aren't totally price representative. Neither include an optical drive as A) they are cheap B) it's likely you probably have one already that might be reusable. Also, if you do need to fit an OS into the budget, you'll be looking at something nearer the $350 build, with probably a bit more spent on graphics.