How to integrate WordPress Core updates with your custom Plugin or Theme

gistfile1.php

<?php
/**
 * How to integrate WordPress Core updates with your custom Plugin or Theme
 * 
 * Filter the `update_plugins` transient to report your plugin as out of date.
 * Themes have a similar transient you can filter.
 */
add_filter( 'site_transient_update_plugins', 'wprp_extend_filter_update_plugins' );
add_filter( 'transient_update_plugins', 'wprp_extend_filter_update_plugins' );
function wprp_extend_filter_update_plugins( $update_plugins ) {

	if ( ! is_object( $update_plugins ) )
			return $update_plugins;

	if ( ! isset( $update_plugins->response ) || ! is_array( $update_plugins->response ) )
		$update_plugins->response = array();

	// Do whatever you need to see if there's a new version of your plugin
	// Your response will need to look something like this if it's out of date:
	$update_plugins->response['wpremote-sample/wpremote-sample.php'] = (object)array(
			'slug'         => 'wpremote-sample', // Whatever you want, as long as it's not on WordPress.org
			'new_version'  => '0.2', // The newest version
			'url'          => 'https://github.com/humanmade/wp-remote-sample-plugin', // Informational
			'package'      => 'https://github.com/humanmade/wp-remote-sample-plugin/archive/0.2.zip', // Where WordPress should pull the ZIP from.
		);

	return $update_plugins;
}