Have a go with this:
http://www.nerdmuch.com/games/1495/best-game-capture-cards-devices-twitch-streaming-elgato-game-capture-hd60-hd-pvr/
Likewise you have brands such as Elgato, Avermedia, Matrox to name a few.
You can have a
read through this thread to help you get a foot up on the matter. You've also figured out that your budget will govern which route you take in your purchase.
Further reading material:
https://www.reddit.com/r/Twitch/comments/2bwsxr/a_general_guide_to_console_streaming_capture/
http://help.twitch.tv/customer/portal/articles/1988680-broadcasting-with-two-computers
https://obsproject.com/forum/resources/q-a-about-dual-pc-stream-audio-setup.156/
If you're looking to drop in a capture card into your system then there are PCIe versions available by the above stated manufacturers
http://www.amazon.com/dp/B014MQIVPS
http://www.newegg.com/Product/Product.aspx?Item=N82E16815100100
but from what I understand one system needs to offload streaming duties.
Hope this helps.