That's not what most consider a budget build. With that much money there's no reason not to go with Intel.
Yeah. That's pretty much it.
However, let's play devil's advocate. Newegg build time! I'll choose exactly what I think would make the best Intel and AMD gaming systems that are well rounded PCs and would provide enjoyment and minimal annoyance. I'll aim at $1000 right with a legal OS, and pick things that have low shipping or free shipping, most importantly for the case). For the sake of streamlining this, we'll assume the person has an existing display/keyboard/mouse/speakers of some type.
Best deal on AMD Gaming build :
Windows 8.1 Pro Keycard : $99 (download ISO from different PC, write to USB or DVD)
Rosewill Redbone Case : $35 (free shipping)
Enermax 550W ECO Modular : $70
DVDRW : $20
Samsung EVO 128GB SSD : $85
Toshiba 2TB HDD : $75
Kingston HyperX 8GB (2x4GB) DDR3-1866 : $70
Cooler Master Hyper 212 Evo : $35
AsRock 980DE AM3+ Mobo : $70 (8 Sata, USB 3.0, Sata 6gps, used these to do 8320 @ 4.7Ghz rock solid before)
AMD FX-6300 CPU : $120 (stock 3.5Ghz, dead easy/silent to hit 4.2Ghz on stock volts)
MSI TwinForce 770 2GB OC Edition : $320
Okay, so that brings us to $1k dead even. Ends up with fast SSD+2TB Storage setup, mobo that supports up to 32GB ram and can OC into the 4.5Ghz range if desired, 6-Core FX under a cooler that is VERY quiet and keeps it chilly, a VERY nice gaming GPU, in a nice case with a rock solid PSU.
Let's shift the things around to fit in an Intel i5.
AsRock Pro4 Mobo $105
Intel 4670K CPU $240
XFX R260X 2GB OC $150
So you end up with the same thing, but a better CPU combined with a 260X (7790 rebadge). For gaming, I'd take the FX @ 4.2Ghz + GTX 770 every time.
Of course, one could cut the SSD, go with a cheaper PSU, try to scrape some dollars here and there. Anything to keep the SSD in the system.
If the price difference was zero I'd pick the i5 100/100 times. With limited funds, it gets more interesting. I guarantee that in the vast majority of games, the 4.2Ghz FX6300 setup + GTX770 would wipe the floor with the i5 + 7790/R260X.