I have a RiftS and use a couple of programs to do it. First, there's Oculus mirror that's part of the Oculus software. You run that while playing AH. It has to be maximized on the monitor. It shows exactly what the headset sees, actually a bit more FOV.
Also use OBS. That's a free software you can use to record and stream. Use the setup wizard to get basic settings set up easy. You will have to set other stuff up. One being Oculus mirror as a window capture source, others being your mic and desktop audio. Another would be some hot keys to start and stop things.
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