AMSTi :
CPU - AMD 8350
RAM - 32GB Avexir 4x8
Motherboard - ASRock 990FX Extreme9
Storage - x3 1TB Samsung 850 Pro SSD
PSU - Corsair AX1500i
Case - Silverstone TJ11B-W
4K monitor - Asus PB287Q
1080P monitors - x5 Asus VN289H
Ah - ok, well at that point the R9-290 would be good to go to run the BF4, that would take one of the PCI-e x16's and leave you 3 pci-e slots and about 16 lanes (x4/x4/x4). After that though you'll need to pick up some other ATI cards which support at least dual monitors.
Actually - the R7-250 will support up to three monitors - this depends though on what kind of connections you have on the monitors, careful shopping could get you the XFX R7-250 Ghost edition which supports dual-link DVI (2 monitors there) plus an HDMI or Displayport connection (monitor 3) and is a single slot solution. Two of those would handle the additional 5 monitors with room for one more. You might go a stitch over budget though by about $50 or so.
Are the monitor connections DVI, VGA, HDMI or display port?