Can someone please help me make this decision on this??

What is your budget for your actual GPU? I wouldn't really recommend either of the choices, especially the 6770 Crossfire being it's 4 yr old GPU technology. Depending on what your looking to spend, you can get much better cards.

You should be fine, but if you want, you could always upgrade to a better graphics card, like the GTX 960 for $230, which is a lot better than the 7870. Or an r9 270x for about $180, which is basically the 7870 on crack.

Do you have your motherboard, case, PSU, ect. already picked out. This way it will be easier to figure out the best GPU option for you.

Asus M5a97 r2.0 or any asus/asrock 970 chipset would be good.

Well, the M5A97 r2.0 costs $90 on newegg.

It's quite nice, decent overclocker, 32gb max ram, UEFI BIOS, windows 8.1 ready, usb 3.0, slot for 2 graphics cards, a good low cost motherboard in general.

Of course it is compatible with previous versions. I got windows 7 64bit running on it, and I had linux on it before.

Yes, but it'll go up to a 8350 at most.

Can I know what you already picked out and how much money you got left to spend?

Well, for the ram, just buy the cheapest ddr3 1600 ram you can find that has good reviews. You'll find them around 60~70 USD, and lower if you're lucky.
As for the case, it really comes down to personal preference. You could sacrifice component power for a better looking case, or just get a shitty $15 case that gets the job done, albeit horribly (poor cable management, bad airflow, high temps, etc...).
The power supply could be any 500+ watt power supply from corsair, evga, or seasonic. Just make sure it it 80+, and not the CX series from corsair. 450W would be okay too, but 500W is about the sweet spot for budget builds.
The hard drive should be Western Digital Black, or Blue if you want to leave more money for the graphics card, however much space you think is good, but I recommend 1TB. If you can save up more money within the next few months, then just get a WD Blue and then buy a 250GB SSD for around $130.
Finally, you got the graphics card. So far the fx 6300 ($90~100), 8GB ram ($60~70), M5a97 ($80~90), hard drive (50~80), power supply ($20~50), case (20~60) will leave you from 180 to 50 dollars on a graphics card with $500 budget. Prices change on a daily basis, so there will most likely be a mix of the prices, so let's assume $130 for the graphics card. For a new graphics card, you should be expecting a gtx 750/ r7 260x type of graphics card. At most, you'll get an r9 270, which you could then overclock a bit to get to the r9 270x level. This set up should be okay, because with the fx 6300 and a gtx 750 tier graphics card, you could run most games on medium to high settings. With the r9 270, you should be maxing out most games except for the most demanding ones ( Crysis 3, BF4, Far Cry4, etc..). On the other hand , if you get a used (refurbished if you want to be safe, but costs slightly more) card, you could find HD 7950's for around 90~140 floating around on the internet (ebay, amazon, craigslist, etc..). This graphics card is slightly worse than a r9 280, but has a pretty high power draw, so you'll have to get atleast a 500W PSU, 600W to be safe. I don't recommend Nvidia, as they usually cost more for the same performance, due to better energy efficiency.

All the graphics cards that I talked about will be able to run gta v atleast on high. Although the fx 6300 might have some minor bottlenecks without overclocking.

I can't; because I didn't give you a build list. I gave you all the options. Now reread what I wrote in the last last comment, and you'll have a rough idea of what I'm telling you. If you still have any questions, you may ask, me. But don't tell others to make a partlist for you, because it is your build, and I wouldn't know what you wanted. What kind of case you want? Would you sacrifice performance to invest more on a better looking case? Would you buy a smaller HDD to get a better gpu, and get an SSD later, or just get a big HDD and a lower gpu? Would you want a cheaper motherboard, or a more expensive one for overclocking? Would you save some money and get a cheap $20 psu that will fail in a few years, or pay more for better quality? Would you buy a new graphics card for the safety of something unused, or would you get a better graphics card that used? These are choices that you have to ask yourself, and no one else can make for you.