GTA V is actually a pretty well optimized game - hitting 60fps on any decently modern machine won't be difficult. I would think you could keep it under $1500, like UgAhgItHurts said.
It also depends on your monitor. I'll assume 1080p? 1080p is fairly easy to push these days.
The main factor will be the graphics card. You can pick up a 290X for around $300, which is a steal. the 300 series will be out toward the end of June, if you want to wait until then, but the 290X is a really good deal right now, and for 1080p, will be more than enough for you. The NVidia equivalent would probably be the 780 ti, or if you want the new generation the 980, but both of those run $400-$500, so it gets more expensive.
The second thing to consider is the CPU - I'll assume you don't want to overclock, as it makes it a lot more complicated and it can get more expensive as well. You could go with something as simple as an i3, or get a Xeon equivalent of an i7 if you really want to, because it will have similar performance without the overclock. If you want to go the AMD route, both the 6300 and the 8350 are popular, and are cheap to overclock as well - you would want to get an aftermarket cooler if that were the case.
Here's a build I made for you. It doesn't account for an operating system, which you'll need (probably Windows), but I got it to within roughly $800, and it should be able to handle GTA V just fine. Second build is around $900 with a more powerful CPU. I also got it to be roughly themed for you The case doesn't really matter as long as it's good. I've heard good things about the S430, but you could choose the H440 (which I have) or the Fractal Design Define R4/5, or any other one, if you want.
$800: http://pcpartpicker.com/p/PQ96Vn
$900: http://pcpartpicker.com/p/nKggxr
If you want any other suggestions (Intel/NVidia build, AMD/AMD build, etc. etc.), feel free to let me know.
(Also, keep in mind you have to get a wifi card if you don't want to or can't use ethernet. I use the Intel 7260, which is rather expensive, but it comes with Bluetooth.)