You could create a service to check this out and warn the user, but I have a better option, usually android devices have to enable automatic update checking, so when you drop a new version on the Play Store it would update itself if was connected to the internet.
Steps to do as I said:
To automatically update apps on your Android device:
Open the Google Play Google Play app.
Tap Menu Menu then Settings.
Touch Auto-update apps.
Select an option:
- Update auto-apps at any time to update apps using cellular or Wi-Fi network data.
- Automatically update apps over Wi-Fi only to update apps only when your device is connected to a Wi-Fi network.
Note: If an account on your device displays a login error, apps will not update automatically.
Take a look at here for more information!