We'll need more info on your gaming build. I wouldn't suggest streaming with a second computer because you'll just waste electricity and there's going to be most likely be sync problems with video/audio. Though i would suggest to have a second monitor because capturing with one monitor sometimes is a pain, it's good to have the second monitor to do all the work while the first monitor streams.
If you have a really good graphics card and a good processor, then i would recommend just use an streaming program like OBS or xsplit and stream out of the card. If that's a problem maybe an aver media capture card would suffice. I can't really give a good suggestion of one or another until we know your computers set up
Well something must be going wrong in bf4...you should have no problem with 2 x 670's FTW edition. Sounds like BF4 is not utilizing SLI which means your second card is just sitting there doing nothing. You should be able to get really good fps with ultra and stream. If that's the case you might need a capture card then.