Simple Android Java class that allows you to determine whether or not to show a "rate my app"-dialog and allows you to show the dialog that takes the user to the Play Store when the dialog is confirmed.

RateAppManager.java

public class RateAppManager {

    private static final String MS_PREFS = Prefs.class.getSimpleName();
    private static SharedPreferences preferences;

    private static final String APP_START_COUNT = "appStartCount";
    private static final String APP_INSTALL_TIMESTAMP = "appInstallTimestamp";
    private static final String RATE_DIALOG_SHOWN = "rateDialogShown";

    private static SharedPreferences getPreferences() {
        if(preferences == null) {
            preferences = MyApplication.getAppContext().getSharedPreferences(MS_PREFS, Context.MODE_PRIVATE);
        }
        return preferences;
    }

    // should be called in the SplashScreen- or MainActivity
    public static boolean shouldShowRateDialog() {
        SharedPreferences prefs = getPreferences();

        // dialog already shown and dismissed?
        if(prefs.getBoolean(RATE_DIALOG_SHOWN, false)) {
            return false;
        }

        int starts = prefs.getInt(APP_START_COUNT, 0);
        Calendar now = Calendar.getInstance();
        
        // app started for the first time, save start date
        if (starts == 0) {
            prefs.edit()
                    .putLong(APP_INSTALL_TIMESTAMP, now.getTimeInMillis())
                    .putInt(APP_START_COUNT, 1)
                    .apply();
            return false;
        }
        // only show dialog after 10 starts ...
        if(starts < 10) {
            prefs.edit()
                    .putInt(APP_START_COUNT, ++starts)
                    .apply();
            return false;
        }
        
        // ...and if the app is installed for more than 3 days
        return now.getTimeInMillis() - prefs.getLong(APP_INSTALL_TIMESTAMP, 0) >= 1000 * 3600 * 24 * 3;
    }

    public static void showAppRateDialog(final Context context) {
        // context has to be an activity context, not the application context!

        // styled
        AlertDialog.Builder builder = new AlertDialog.Builder(new ContextThemeWrapper(context, R.style.AlertDialogCustom));
        // default style
//        AlertDialog.Builder builder = new AlertDialog.Builder(context);

        builder.setTitle(context.getString(R.string.rate_app_title));
        builder.setMessage(context.getString(R.string.rate_app_message));
        builder.setNegativeButton(context.getString(R.string.rate_app_maybe_later), new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                AppRateManager.postponeRateDialog();
                dialog.dismiss();
            }
        });
        builder.setNeutralButton(R.string.rate_app_no_thanks, new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                AppRateManager.rateDialogShown();
                dialog.dismiss();
            }
        });
        builder.setPositiveButton(R.string.rate_app_yes, new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                AppRateManager.rateDialogShown();
                dialog.dismiss();
                Uri uri = Uri.parse("market://details?id=" + context.getPackageName());
                Intent goToMarket = new Intent(Intent.ACTION_VIEW, uri);
                try {
                    context.startActivity(goToMarket);
                } catch (ActivityNotFoundException e) {
                    context.startActivity(new Intent(Intent.ACTION_VIEW, Uri.parse("http://play.google.com/store/apps/details?id=" + context.getPackageName())));
                }
            }
        });
        AlertDialog dialog = builder.create();
        dialog.show();
    }

    private static void rateDialogShown() {
        getPreferences().edit().putBoolean(RATE_DIALOG_SHOWN, true).apply();
    }

    private static void postponeRateDialog() {
        // show again in two days by setting the install date to one day ago
        Calendar now = Calendar.getInstance();
        long oneDayAgo = now.getTimeInMillis() - 1000*3600*24;
        getPreferences().edit().putLong(APP_INSTALL_TIMESTAMP, oneDayAgo).apply();
    }
}