I've used the in-home streaming, it's actually pretty nifty. For a test run (when I first saw it on my account), I used my Win8 tablet and it worked pretty well. Minimal input lag too and I was playing F1 2013, so something that requires you to react pretty quickly. And it is cross-OS compatible as I tried it again on my Mac Mini.
So for this build, you can certainly get away with as cheap as possible:PCPartPicker part list
/ Price breakdown by merchantCPU: AMD 5150 1.6GHz Quad-Core Processor
($49.99 @ SuperBiiz) Motherboard: MSI AM1I Mini ITX AM1 Motherboard
($33.99 @ Amazon) Memory: Crucial 4GB (2 x 2GB) DDR3-1600 Memory
($44.99 @ Newegg) Storage: Western Digital WD Blue 320GB 2.5" 5400RPM Internal Hard Drive
($39.99 @ Amazon) Case: Antec ISK 110 VESA Mini ITX Desktop Case w/90W Power Supply
($62.20 @ Amazon) Total:
$231.16Prices include shipping, taxes, and discounts when availableGenerated by PCPartPicker 2014-09-14 16:10 EDT-0400
The case comes with a PSU and it's small enough to be VESA mounted.
However there is a caveat to streaming: the computer who's hosting it is running the game as if you were there. So local co-op will need to be played as if you were playing on that machine (i.e., split screen or otherwise).