I'm building a HTML5 Game and deploying it to my Android phone for testing using Apache Cordova.
Is there anyway to debug my application on the device while it's running, such as seeing the JS Console?
Have you taken a look at the debug.phonegap.com?
Another way you can upload the html to a server or localhost and use it. But this only for outside the apk:
https://developers.google.com/chrome-developer-tools/docs/remote-debugging