La clase Android LocationClient está obsoleta pero se usa en la documentación

Parte de la documentación es antigua en Google (algunos ejemplos que mencionaste todavía usan el obsoleto LocationClient). Tienes que usar el nuevo GoogleApiClient como se describe en los ejemplos de Servicios de localización:

private GoogleApiClient mGoogleApiClient;

  mGoogleApiClient = new GoogleApiClient.Builder(context)
     .addApi(LocationServices.API)
     .addConnectionCallbacks(this)
     .addOnConnectionFailedListener(this)
     .build()

Y cuando el nuevo cliente está conectado, puede usar la api de ubicación fusionada, por ejemplo, de la siguiente manera:

LocationServices.FusedLocationApi.requestLocationUpdates(theNewClient, 
    locationRequest, locationListener);

Parece que esto fue cubierto en el blog de desarrolladores . Para LocationClient, usarías esto junto con LocationServicesque luego nos lleva a GeofencingApi.

Se elimina LocationClient. GoogleApiClient es una api que se puede usar para las API de ubicación de Google Play Services.

El código de ejemplo para los escenarios comunes es aquí y las clases de entrenamiento se actualizaron con más próximamente.

De acuerdo con el código de actualización de la documentación ..

package iwannado.com.myapplicationforsha1key;

import android.Manifest;
import android.content.Intent;
import android.content.pm.PackageManager;
import android.location.Geocoder;
import android.location.Location;
import android.support.annotation.NonNull;
import android.support.annotation.Nullable;
import android.support.v4.app.ActivityCompat;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.util.Log;
import android.view.Menu;
import android.view.MenuItem;
import android.view.View;
import android.widget.EditText;
import android.widget.Toast;

import com.google.android.gms.common.ConnectionResult;
import com.google.android.gms.common.api.GoogleApiClient;
import com.google.android.gms.identity.intents.Address;
import com.google.android.gms.location.LocationListener;
import com.google.android.gms.location.LocationRequest;
import com.google.android.gms.location.LocationServices;
import com.google.android.gms.maps.CameraUpdate;
import com.google.android.gms.maps.CameraUpdateFactory;
import com.google.android.gms.maps.GoogleMap;
import com.google.android.gms.maps.MapView;
import com.google.android.gms.maps.OnMapReadyCallback;
import com.google.android.gms.maps.model.CameraPosition;
import com.google.android.gms.maps.model.LatLng;
import com.google.android.gms.maps.model.MarkerOptions;

import java.io.IOException;
import java.util.List;

public class MapWithMapViewActivity extends AppCompatActivity implements OnMapReadyCallback,
        GoogleApiClient.ConnectionCallbacks, GoogleApiClient.OnConnectionFailedListener, LocationListener {
    private static final String TAG = MapWithMapViewActivity.class.getCanonicalName();


    private GoogleMap mMap = null;
    private MapView mMapView = null;

    private EditText loatcationEditText = null;

    private GoogleApiClient mGoogleApiClient = null;
    private Location mLocationRequest = null;


    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_map_with_map_view);
        loatcationEditText = (EditText) findViewById(R.id.loatcation_edit_text);
        mMapView = (MapView) findViewById(R.id.mapView);
        /*
        * The method is used to create mapView
        * */
        mMapView.onCreate(savedInstanceState);
        /*
        *The method Return Google map
        * */
        mMapView.getMapAsync(this);

        gotoCurrentLoactionGooglePlayService();
    }

    @Override
    protected void onResume() {
        super.onResume();
        mMapView.onResume();
    }


    @Override
    protected void onPause() {
        super.onPause();
        mMapView.onPause();
    }

    @Override
    public void onLowMemory() {
        super.onLowMemory();
        mMapView.onLowMemory();
    }

    @Override
    protected void onStart() {
        super.onStart();
        mGoogleApiClient.connect();
    }

    @Override
    protected void onStop() {
        super.onStop();
        mGoogleApiClient.disconnect();
    }

    @Override
    protected void onSaveInstanceState(Bundle outState) {
        super.onSaveInstanceState(outState);
        mMapView.onSaveInstanceState(outState);
    }

    @Override
    protected void onDestroy() {
        super.onDestroy();
        mMapView.onDestroy();
    }


    @Override
    public void onMapReady(GoogleMap googleMap) {
        mMap = googleMap;



    }



    private void gotoCurrentLoactionGooglePlayService() {
        /*working with new google api for laction..
http://stackoverflow.com/a/25173057
* */
        mGoogleApiClient = new GoogleApiClient.Builder(this)
                .addApi(LocationServices.API)
                .addConnectionCallbacks(this)
                .addOnConnectionFailedListener(this)
                .build();
    }



    @Override
    public void onConnected(@Nullable Bundle bundle) {
/*
* Follow this documentation.. https://developer.android.com/training/location/retrieve-current.html
*
* Please add Runtime permission here for android 6
* */
        mLocationRequest = LocationServices.FusedLocationApi.getLastLocation(
                mGoogleApiClient);
        if (mLocationRequest != null) {

           LatLng latLng = new LatLng(mLocationRequest.getLatitude(),mLocationRequest.getLongitude());
        mMap.addMarker(new MarkerOptions().position(latLng).title("Programmatically Current Loaction"));
        mMap.moveCamera(CameraUpdateFactory.newLatLng(latLng));
            Toast.makeText(this,"getLatitude() = "+String.valueOf(mLocationRequest.getLatitude())+"\n getLongitude() = "+String.valueOf(mLocationRequest.getLongitude()),Toast.LENGTH_LONG).show();

        }
    }


    @Override
    public void onConnectionSuspended(int i) {
        Log.i(TAG, "GoogleApiClient connection has been suspend");
    }

    @Override
    public void onConnectionFailed(@NonNull ConnectionResult connectionResult) {
        Log.i(TAG, "GoogleApiClient connection has failed");
    }

    @Override
    public void onLocationChanged(Location location) {


        Toast.makeText(this,"Location received: " + location.toString(),Toast.LENGTH_LONG).show();

    }
}