How can I get the apk file of an app?
How can I get the APK file of an app after installing it from Google Play? 


Multiple ways, but the easiest one, I suppose, is to install a backup type app such as App Backup Restore. Take the backup of the the target app and its APK would be saved inside a directory under the default storage (/sdcard).


  Once I find the APK file of an app, can I copy it out of my android phone to my Ubuntu laptop and then extract the hidden document file in the apk file? Or do I have to extract on my android phone?


An APK is an archive (MIME type: application/zip). You can copy it to your laptop alike any multimedia file. 

As for the extraction of the contents, there are multiple apps on Play Store which supports extracting a ZIP file, so it is up to you to extract the app on Android or in your PC to explore its contents.