How can I download an APK file from the Play Store? I can't download apps from the Play Store on my Android device (maybe due to a network issue), so I'm wondering how could I download the APK file of an app on my PC from http://play.google.com/store?
I spent a decent amount of time trying to figure this out because I don't feel like having the "google experience" and as far as I can tell, it isn't possible. I've worked around it:

The F-Droid repository and Amazon Appstore are both reasonably well maintained -- either will manage any apps you install, which is handy.  That's where I start.

Otherwise, I track down and download the .apk outside of the google-run market. In many cases the Android market will point you to a developer's web page and contact information -- if they don't publish the .apk on their web page, you can always ask for it. 

Two answers worth checking out:

Where can I find open source Android apps?

What are the alternative Android app markets?

PS. Don't forget: to install apps from .apk you have to turn on unknown sources (Settings &gt; Applications &gt; Unknown sources). 

PPS. Once you have the .apk on your computer you know you need to turn on USB Debugging so you can copy the .apk to your phone, right? Settings &gt; Applications &gt; Development &gt; USB debugging