How can I customize permission dialog in Android?

The short answer is, you can't.

As android documentation puts:

When your app calls requestPermissions(), the system shows a standard dialog box to the user. Your app cannot configure or alter that dialog box. If you need to provide any information or explanation to the user, you should do that before you call requestPermissions(), as described in "Explain why the app needs permissions".

So, there is no way to define a custom layout for the permission dialog for now.

You can find more detailed information here: http://developer.android.com/training/permissions/requesting.html

I have create custom dialog for permission:

if (ContextCompat.checkSelfPermission(context, Manifest.permission.READ_EXTERNAL_STORAGE) != PackageManager.PERMISSION_GRANTED) {
   if (ActivityCompat.shouldShowRequestPermissionRationale((Activity) context, Manifest.permission.READ_EXTERNAL_STORAGE)) {
       AlertDialog.Builder alertBuilder = new AlertDialog.Builder(context);
       alertBuilder.setCancelable(true);
       alertBuilder.setTitle("Permission necessary");
       alertBuilder.setMessage("External storage permission is necessary");
       alertBuilder.setPositiveButton(android.R.string.yes, new DialogInterface.OnClickListener() {

       @TargetApi(Build.VERSION_CODES.JELLY_BEAN)
       public void onClick(DialogInterface dialog, int which) {
           ActivityCompat.requestPermissions((Activity) context, new String[]{Manifest.permission.READ_EXTERNAL_STORAGE}, MY_PERMISSIONS_REQUEST_READ_EXTERNAL_STORAGE);}});

       AlertDialog alert = alertBuilder.create();
       alert.show();
   } else { 
          ActivityCompat.requestPermissions((Activity) context, new String[]{Manifest.permission.READ_EXTERNAL_STORAGE}, MY_PERMISSIONS_REQUEST_READ_EXTERNAL_STORAGE);}
    return false;} 
 else { 
     return true;
}

Hope this will help you :)