According to the APK Downloader Chrome extension Guide on the page to which you linked, the extension requires a Google Services Framework Android Device ID. Presumably (based on the red warning, “Do not type in random email or device ID, it won't work”) these have to be a real GSF ID and corresponding e-mail registered with Google Play. If one does not have any Android device other than a Kindle, how might one get such an ID/e-mail pair?

Also, the FAQ on that page claims that paid apps can’t be downloaded. I gather the free version is just a trial, so that would be of no use.

How did you access the paid version for your BlackBerry? (There’s no Google Play for BlackBerry, is there?)