How to update an apk without uninstalling the current apk?

You can use this:

protected String doInBackground(String... sUrl) {
    String path = "/sdcard/YourApp.apk";
    try {
        URL url = new URL(sUrl[0]);
        URLConnection connection = url.openConnection();
        connection.connect();

        int fileLength = connection.getContentLength();

        // download the file
        InputStream input = new BufferedInputStream(url.openStream());
        OutputStream output = new FileOutputStream(path);

        byte data[] = new byte[1024];
        long total = 0;
        int count;
        while ((count = input.read(data)) != -1) {
            total += count;
            publishProgress((int) (total * 100 / fileLength));
            output.write(data, 0, count);
        }

        output.flush();
        output.close();
        input.close();
    } catch (Exception e) {
        Log.e("YourApp", "Well that didn't work out so well...");
        Log.e("YourApp", e.getMessage());
    }
    return path;
}

// begin the installation by opening the resulting file
@Override
protected void onPostExecute(String path) {
    Intent i = new Intent();
    i.setAction(Intent.ACTION_VIEW);
    i.setDataAndType(Uri.fromFile(new File(path)), "application/vnd.android.package-archive" );
    Log.d("Lofting", "About to install new .apk");
    this.context.startActivity(i);
}