It's hard to recommend a good build for a gaming PC on low budget. If it's possible to get used parts which you know are good and you can trust the seller, its always better than full priced products. Try getting sandy bridge, ivy bridge or even haswell and a compatible good quality mobo.
BF4 favors intel CPU's a lot. i5-2500k (sandy), i5-3570k (ivy) or i5-4670k (haswell).
Memory is not that important even though especially ivy bridge's and haswell's favor high frequency sticks. 8 gigs of 1600mhz cl7-9 is good enough for gaming.
For gpu I would go nvidia for gaming, but it is your decision. GTX 760's and 770's are very cheap right now for their performance. You can also optionally pick little bit weaker used cards. (670-680 or equal in performance from AMD, although keep in mind for BF4 the performance will suffer).
This build advice is also meant for a little bit lower graphical quality, 1080p, high fps and good monitor panels that can do 144hz because that is the only way FPS-games are really enjoyable.