I have this build http://pcpartpicker.com/user/ChillinSince96/saved/dwxRsY
Which GPU should I get I play battlefield 4, splinter cell blacklist, skyrim, bf3, and chivalry
Also do AMD Gpu's work better with AMD Cpu's?
It is a wash. One does better in some games the other better in some games.
Generally they will be similar with the 280X being slightly faster and usually cheaper.
No. It doesn't matter what brand your CPU is. You can mix and match. AMD/nVIDIA, Intel/AMD, AMD/AMD, Intel/nVidia. They will work the same.
And if you haven't ordered the parts yet save your money and get the FX-6300 and the 840 Evo.
CPU should make no difference
The 280X is about 20% faster than the 770
that build looks fine - dont forget a CPU cooler
Head to head, they are both about the same performance. It all comes down to how the game is encoded and the code used. Vandals 20% difference has the decimal in the wrong place, there is only about a 2% difference between most games on these two cards in real world testing of the reference cards, not enough to be noticeable. You would be happy with either one.
EDIT
I just noticed the 770 you stated was "superclocked", giving it a little more performance over the reference cards, and a little jump over a reference 280, honestly, I would just go with whichever has the best price.