I don't know a whole lot about the AMD processors any more.
I know the Intel line has been immensely stronger for several years.

Playing with an i5 2500K, was smooth 5-boxing everywhere except SW/Org and mass battles in AV/IoC.
It was definitely playable everywhere.
Adding in streaming software, and it was not enjoyable in SW/Org/AV/IoC.

Playing with an i7 2600K (essentially same as i5 2500K, except 4 logical cores to go with the 4 physical cores) and everything was smooth everywhere, whether streaming or not.

I don't think AMD currently has a processor as strong (for boxing) as either of these.
But that might give you an idea on the processor end.




For ram, I generally like to have ([1GB for the OS] plus [1GB per instance of Warcraft]).
My preference would be 6GB+ for 5-boxing and 11GB+ for 10-boxing.

Ram is a cheap upgrade, that would help with your endeavors.




According to the Tom's Hardware Video Hierarchy chart, a 7850HD is comparable to a GTX 480.

I would say, 5-boxing with DX9 would be quite possible, but I would be surprised if you could 10-box on that card.
How much video ram does your 7850 have?