The easiest way to debug WebView is to connect your Android device to PC by USB and inspect your WebView by Chrome dev tools.
So, you will need:
1) Activate USB debugging on our devise. You can find it
Settings >> Developer Options >> Debugging >> USB Debugging (activate checkbox)
2) Connect you devise to Computer by USB
Note: If you are developing on Windows, install the appropriate USB driver for your device. See OEM USB Drivers on the Android Developers' site.
3) Open Chrome browser and type in the URL field:
4) Confirm that 'Discover USB devices' activated
5) On your device, an alert prompts you to allow USB debugging from your computer. Tap OK.
6) On the chrome://inspect page displays every connected device. Click
inspect for connected device and you will get console.
More detailed manual is Debugging Android WebView
If you are not using Android 4.4, according to http://developer.android.com/guide/webapps/debugging.html you're going to enjoy the old-school way of debugging ...
For Android 4.4 and better, you'll enjoy remote debugging, see https://developers.google.com/chrome-developer-tools/docs/remote-debugging#debugging-webviews
Starting Android 4.4 (KitKat), you can use the DevTools to debug the contents of Android WebViews inside native Android applications.