I did it but I'd like better. I'm always on a real tight budget.
I started with an Asus P8H77V-LE mobo / i7 2600 CPU combo, used on ebay - I picked that up for £115 (yes pounds).
I found a Zotac GTX 760 Amp edition GPU for £75 (bargain) and went budget on the case.
The GPU will do high to very high settings but not at 60fps. Mostly 40 - 50 which is still 50% faster than my Xbox One.
I got a new PSU with bronze rating and an SSD. All in all I paid about £450.
Pounds to dollars - £450 is around $630, so your $500 budget can be done but you have some serious bartering to do and you'll be looking for used hardware.
I do have GTA and have tried some mods but I could do with a better GPU now. It wouldn't manage graphical mods for sure.
In fact after installing mods I turned the graphic settings down a bit.