Google Play Games Plugin for Unity:Tips & Tricks (http://diegogiacomelli.com.br/google-play-games-plugin-for-unity-tips-and-tricks)

AchievementsController.cs

using GooglePlayGames;
using GooglePlayGames.BasicApi;
using UnityEngine;

public class AchievementsController : MonoBehaviour
{
    void Start()
    {
        var config = new PlayGamesClientConfiguration.Builder().Build();
        PlayGamesPlatform.InitializeInstance(config);
        PlayGamesPlatform.DebugLogEnabled = true;
        PlayGamesPlatform.Activate();

        // Authenticate and register a ProcessAuthentication callback
        // This call needs to be made before we can proceed to other calls in the Social API
        Social.localUser.Authenticate(ProcessAuthentication);
    }

    // This function gets called when Authenticate completes
    // Note that if the operation is successful, Social.localUser will contain data from the server. 
    void ProcessAuthentication(bool success)
    {
        if (success)
        {
            Debug.Log("Authenticated.");
            PlayGamesPlatform.Instance.ShowAchievementsUI();
        }
        else
            Debug.Log("Failed to authenticate");   
    }
}