UPDATE: See below for how to put video on your profile with Box.net.
The process is simple: use Google Docs to create a presentation and embed a YouTube video in the presentation. Then install the Google Presentation app on LinkedIn and select that presentation to be displayed on your profile. With this solution, people can watch YouTube videos right on your profile. If you prefer to have a list of video files they can download, or a video not on YouTube, you can use the Box.net application to put a fileshare folder on your profile too.
- Install the Box.net app on LinkedIn
- Click on the folder labeled "Profile folder"
- Click on the blue arrow and select "Upload"
- Select a .flv/Flash video file from your computer
Once it's uploaded, anyone can view the video from your profile.