When you are watching videos on YouTube, there are two ways to promote your book, one that is paid and one that is technically paid but practically free or at least very cheap.
For either method, you’ll need to signup for AdWords for YouTube as that is the program that is used.
The paid method is to purchase ads on specific videos (or keywords that cause a video to show up) and it will work similar to your standard AdWords campaign where you pay per click.
The semi-cheap method is to create a Promoted Video campaign through the AdWords interface in YouTube so that people will watch your video. When folks search for keywords, there are normal listings and promoted listings, and you need to get your video into the promoted listings. Now, you don’t have to actually pay for much in order to have your video be promoted; set a penny a day limit and choose an obscure keyword so that your video doesn’t really show up if you want to make it unlikely you’ll have to spend that penny, or use the system as it’s designed and make an accurate ad that actually promotes your video. It could be a good return on investment if people then buy your book…
Once a video is being actively promoted at some level, it unlocks the Call to Action Overlays feature on that video. You can then set up a free ad on your own video that appears and you can use that to link to your book. (Technically, you are paying for folks to watch your video, and the CTA overlay is what you are trying to get them to click on.)
YouTube provides Step by Step Directions here for how to use that feature, so check there for the most up to date instructions.