Its an odd question, mainly because you will be limiting your own gaming experience I guess.
Will you be content will gaming 1080p @ 60fps?
I ask this because mainstream capture cards have only recently given you the ability to capture up to 1080@60fps.
**You can capture higher resolutions but you will still be capped to either 30 or 60fps and you will most likely need to convert from HDMI to SDI.
If you are happy with 1080p@60fps then the specs for a PC won't be that bad.
You'll need a capture card: Avermedia Extreme Cap U3 (USB), Elgato HD60 (USB), Blackmagic Decklink 4k (PCIe). (other options available for 1080i / 720p)
But other than that an i5 / 8GB will be plenty. The Elgato capture card compresses the data stream for you, so the CPU has less work to do, but the Avermedia & BM-Decklink 4k will capture RAW video, so you rig will need more (moar) power to compress for the stream.
Even though you capture in 1080p, my guess if that you won't stream @ this resolution, as Twitch only allows partners to let their viewers to chose their stream quality. And UK internet blows. But it does mean that your local copy will be decent for Youtube if you produce content for that.
Also local direct recording with RAW video will need some RAID 0 HDDs / SSDs.
Personally I would only use this method for console streaming.
££ would be better spent beefing up your current rig and streaming from that.