Hello everyone and thank you for this post,
I've used Google Play Services to handle achievements and leaderboards in my game, and thanks to this post I managed to do it quickly.
I just want to add a few more information: I've been able to use the Xamarin NuGet package (Xamarin.GooglePlayServices.Games v.42.1021.1) in addition with MonoGame v3.6, but I needed to update the Android target version to Android 7.0 (API 24).
Here is a working example provided by Xamarin and specifically made for achievements and leaderboard.
Please note that if you use AppVeyor to make your Android build, you need to manually update Android SDK. Here is the script that I use, and here how to use it.
It's not as easy as it sounds to make it works as expected, especially for this story about debug keystore, and I planned to write an article to explain the process in detail, I will do it for sure as soon as I have some time.