To record a good quality video of your Linux desktop, there are many ways to do this. The ffmpeg utility will be a natural choice for what we need. Run the command below to start recording a video.
localhost% ffmpeg -y -video_size 1600x900 -framerate 30 -f x11grab -i $DISPLAY+0,0 -c:v libx264 -qp 0 -preset ultrafast base.mp4
Just change the command to suit your desktop resolution. Then run this command to encode the video to the webm format for upload to a website.
localhost% ffmpeg -i base2.mp4 -c:v libvpx -qmin 0 -qmax 50 -crf 10 -b:v 2M output.webm
This may then be uploaded to the web. If it is embedded in a website, then Firefox can play it easily.
Here is a sample recording of my Gnome desktop.
Here is more information on recording webm files on Linux.