No, you cannot use Chromecast with Amazon tablets. Chromecast requires Google play services. Kindle Fire or Fire tables do not have this.
Some Kindle Fire or Fire tablets has screen mirroring feature. It uses Miracast. If your TV supports Miracast, you can directly mirror the screen to TV without using any dongles. If your TV does not support Miracast, you can get a Miracast dongle for the TV. Please read this guide on this: https://allaboutchromecast.com/chromecast-vs-miracast-everything-you-need-know-about-wireless-display-and-screen-mirroring/