If the updated apk has the same package name and is signed with the same key you can just send a intent which will call a default android installer. The installed apk will be overriden.
Intent intent = new Intent(Intent.ACTION_VIEW);
Uri uri = Uri.fromFile(new File(pathToApk));
intent.setDataAndType(uri, "application/vnd.android.package-archive");
startActivity(intent);