I cannot be asked to build a whole PC for you as these guys can,
However I can say that for a $1000 build, an R9 280x 3GB can get you BF4 on Maximum settings with no compromise.
I have a build which also cost about that price, and have chosen the same card.
It also happens that the processor listed in the second build (above this post) is also the one I have chosen.
Best of luck with your build, you should be able to max BF4 and have a great experience with either of those builds. The second one is rather similar to mine.