This is a guide for how to get a $33 HDMI capture device and H264 encoder that can do 1080p 30 fps or 720p 60 fps. It will only output a H264 encoded stream and is not able to capture the raw HDMI data.
This is just a quick guide based on all the hard work and info from danman's blog post.
Go to ebay, aliexpress or similar and find a LKV373A sender. It is sold as a device for extending HDMI over ethernet cable. You only need to buy the sender, not the pair of sender and receiver. The sender is actually an HDMI capture and encoder device. It should cost ~$33 shipped.