Unless you have a modern card that supports streaming/recording natively, your best bet is probably OBS (and even then, the quality of OBS is often better than that of the native systems). Even better, it's free!
It is a little fiddly to get into, but I can record on my 3-4 year old machine (i5-2500K and GTX570) at 1080p/30fps/7000kbps (game is running at 60+ - I haven't bothered checking specifically) without any issues, and stream at 720p/30fps/2000kbps (this is downscaled, native is still 1080p), again without any issues.
The thing to remember is to use game capture, not screen capture (at least for OBS).