Ok, probably a $500 PC could run Minecraft with 60FPS minimum without an issue but Battlefield at max settings with 60FPS minimum? Not a chance.
You may want to spend at least $600 to run it on low to medium, $800 to run it on high to ultra and maybe 1000$ to run it smoothly on max settings.
You also won't be able to find a PC on the internet that could run BF4 on ultra that costs less than $600, you'll have to build one if you want to at least run it on low.
Now, if you don't want to spend more than $500, then this is the best you are going to get:
http://pcpartpicker.com/p/BC3GnQ
Don't expect BF4 to run on low with more than 40FPS, maybe +50FPS on the lowest resolution.
And I guess you don't really want to play that game like that right?
Like I said, $600 may be a little bit better, you may be able to run BF4 a little more smoothly on low, maybe even on medium or possibly high. This is the rig:
http://pcpartpicker.com/p/L9sQf7
And if you want to run it on ultra, at least with more than 30FPS, this is the one:
http://pcpartpicker.com/p/q7krYJ
Maybe not 30FPS on ultra, probably more like a 60FPS on ultra, maybe 50FPS maxed out because of the killer card, but don't get this build, the PSU would probably die in 30 minutes after the first time you turn it on.
And if you want to go full ultra on games like Crysis 3, Battlefield series plus many others with 60FPS, go for this one: http://pcpartpicker.com/p/wL6bCJ
Anyways, so if you want to play BF4 on ultra, you'll probably want to spend more than $500, at least $700 if you want a good experience on medium or low.
If you need anything else, be sure to update your answer