Download App Directly to PC - Help

I have a Kindle Fire. It is my only Android device. I can "sideload" android apps to the Kindle by downloading an app directly to my PC and then transfering it to the Kindle. There was a time when I could do this.

However, now it seems I cannot download without sysnching up to a supported Android device (Kindle is not supported from what I can tell). Is there a work-around? How can a get the "apk" files in a direct fashion? Walk me through an example app (WeWe Bridge would be good) if you can.

Please Help! Thanks.