It has always been difficult to watch Amazon Instant Video on Android devices due to Amazon not providing an app to do so. Now you are able to, but it will take a few steps to get there.
First, ensure that you are able to install applications from Unknown sources. This can be done by simply going to your device settings, tapping 'Security', and flipping the switch for 'Unknown sources'. This will allow you to install applications that were not downloaded directly through the Play Store.
Second, download the latest version of Amazon Instant Video from APKMirror at the following link from your device:
http://www.apkmirror.com/apk/amazon-mobile-llc/instant-video/
Once this has finished downloading, simply click the notification and then choose 'Install' on the dialog that pops up. It will take a moment for the app to install, but once it is done you will be able to log in to your Amazon account and watch Amazon Instant Video!
Third, be sure to go back into your 'Security' settings and turn 'Unknown sources' back off so that you do not accidentally install an app that you did not mean to install.