It is annoying that Amazon Instant Video doesn't directly have an option to cast to a Chromecast. You can, however, configure[0] Amazon Instant Video to use Flash instead of Silverlight. Once you switch to using Flash, you can use the "Cast Tab" feature of the Google Cast extension to watch your video on a Chromecast, assuming you're using Chrome.
[0] https://www.amazon.com/gp/video/settings (near the bottom of the page)