My opinion?
Don't waste your time with quantity over quality. You'll get some cheap thrills from the surround effect for a couple minutes, and then you'll probably just continue on being un-amazed by all the speakers surrounding you (while the front ones anchor 90% of everything anyways).
Two good speakers will beat seven dirt cheap ones all day, every day.
If I had 800 dollars, I'd probably get:
two of these:
JBL LSR305 5" Active Studio Monitor | Sweetwater.com
At the buy one-get one half off if you buy before Sept 1st, that's < $225 for a pair of acclaimed powered studio monitors.
And one of these receivers:
MARANTZ SR5008 7.2 Network Home Theater Receiver with AirPlay | Accessories4less <-- I know it's not the cheapest receiver out there but it has preamp outputs which I find to encourage high flexibility. You're gonna need pre-outs for powered monitors.
And you're gonna be under 800 and leaving yourself open for both future upgrades (surround sound + subwoofers) and current pleasure.
I'm a gamer too, and I've got a full 7.1 system if you wanna get technical. But I'd take a solid 2.0 all day, any day over the Onkyo HTiBs you're looking at. As a rule, if you can get it all in a box, there's some serious compromises being made.
Also, If you're willing to go $75 over your budget, the LSR308s (larger, more powerful version of the above linked LSR305s, same website, same buy one get one half off deal before Sept 1) are probably a no-brainer. Two of those quality, tuned-for-pros 8 inch midwoofers are going to deliver much more accurate bass than any cheapo HTIB sub.
Actually, if the system's only really gonna connect to your PC, I think skipping a receiver for now, grabbing two LSR308s alone,
http://www.sweetwater.com/store/detail/LSR308
And one of these:
http://hifimediy.com/sabre-dac-uae23
and one of these:
https://emotiva.com/products/electronics/control-freak
Willl deliver incredible results and pass
well under budget.
I know you're probably disappointed. You just want approval on the magical 7.1 for $800. I get it. I'm sorry. But a great 2.0 will truly impress you and give you the foundation for expanding to 5.1 or 7.2 or whatever in my opinion, whereas a cheap HTiB will just be a set-it-and-forget it thing that will probably rarely if ever ::impress:: you.