Your current one is plenty fine for that depending on what game you play, but since this is in your budget you might as well.
AMD FX 8350 Black Edition 8 Core Processor 4.00GHz Piledriver Socket AM3+ - FD8350FRHKBOX - Scan.co.uk
As to the GPU, yes that is true to an extent, but you can't go out and buy a 4GB GT 640 thinking it'll play BF4 on ultra because it has a high amount of VRAM. That is marketing BS at its finest. My GTX 580 only has 1.5GB of VRAM and I play games fine because when it came out it was a beast of a card (between GTX 660 and 660ti in performance now). I'm the first to admit it needs more VRAM though.
On the topic of things, my point still stands. The 270x isn't much of an upgrade over the 7850 in terms of raw performance. I would say get your new CPU and see how that fairs for you and if you still need a better GPU save and get the 280x at the minimum or a GTX 760 at least.
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