public class UltimoPaso extends AppCompatActivity implements ActivityCompat.OnRequestPermissionsResultCallback {

private FusedLocationProviderClient client;

protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);

client = LocationServices.getFusedLocationProviderClient(UltimoPaso.this);

.....

btnok_ultimo.setOnClickListener(new View.OnClickListener() {

if (ActivityCompat.checkSelfPermission(UltimoPaso.this, ACCESS_FINE_LOCATION) != PackageManager.PERMISSION_GRANTED && ActivityCompat.checkSelfPermission(UltimoPaso.this, Manifest.permission.ACCESS_COARSE_LOCATION) != PackageManager.PERMISSION_GRANTED) {
    return;
}
   requestPermission();
                        client.getLastLocation().addOnSuccessListener(UltimoPaso.this, new OnSuccessListener<Location>()
{
    @Override
    public void onSuccess(Location location) {
       if(location != null){
          latitude_string = (String.valueOf(location.getLatitude()));
          longitude_string = (String.valueOf(location.getLongitude()));

          Toast.makeText(getApplicationContext(), "DATOS: " + location.toString(), Toast.LENGTH_LONG).show();
       }
     }
});

.... despues del create

private void requestPermission(){
        int MY_PERMISSIONS_REQUEST_READ_CONTACTS = 1 ;
        if (ActivityCompat.checkSelfPermission(this,
                Manifest.permission.ACCESS_FINE_LOCATION)
                != PackageManager.PERMISSION_GRANTED && ActivityCompat.checkSelfPermission(this, Manifest.permission.ACCESS_COARSE_LOCATION)
                != PackageManager.PERMISSION_GRANTED) {

            ActivityCompat.requestPermissions(UltimoPaso.this,
                    new String[]{Manifest.permission.ACCESS_FINE_LOCATION},
                    MY_PERMISSIONS_REQUEST_READ_CONTACTS);
            return;
        }
    }