How much of your current PC can be reused?
$800 might not be enough to build a new PC capable of streaming on ultra settings, depending on how much of your current PC you can reuse. Also, there's not much point in getting 2 SSDs, get one for your OS/Games, then get a mechanical for storage if you need it (the price/performance comes out in favor of mechanical drives for storage still).
If you want to stream, you'll want to get an i7, which is going to be most of your budget. $800 isn't going to leave much room for anything other than a mobo, ram, and GPU. If you need monitor, mouse/keyboard/case/PSU/hard drive, you're looking at probably $1200 at least for a streaming PC. Possibly more if you want a higher end video card.
Here's what's possible with ~$800:
http://pcpartpicker.com/p/KQFst6
does not include mouse, keyboard, or monitor.