How do I add a custom app icon for AppleScript app?

From the Finder. Right click on your Application and go to "Show Package Contents". The app will open up like a folder, navigate down to Contents/Resources/. In the Resources folder there is an .icns file. Replace that file with your custom icon (it will need to be formatted as an .icns file, not just a jpeg or png). I believe it needs to have the same name as the original file (so if it was droplet.icns, that is what you should name your replacement .icns file). Now you can close the folder, and do a get info on the App. You should now see the replacement icon for the App.

If it doesn't show up in the finder with the App, while in the Get Info page, select the icon in the upper left hand corner, copy it and paste it, that should update the finder icon with the new icon.