Get wp_logout_url function to work on external (non-WordPress) page

To allow WordPress functions to operate outside of the WordPress environment, you need to let your external pages know about and have access to the WordPress Core and API environment.

http://codex.wordpress.org/Integrating_WordPress_with_Your_Website

The above link is one such reference which covers this very purpose.

Specifically,

In order to transform regular PHP pages into ones that utilize WordPress, you need to add either of the following code snippets to the start of each page.

<?php 
/* Short and sweet */
define('WP_USE_THEMES', false);
require('./wp-blog-header.php');
?>

or

<?php
require('/the/path/to/your/wp-blog-header.php');
?>

Getting your paths right is very important, without, you’ll have no joy.

Refer to the link provided for even greater detail. Any problems thereafter please let us know along with what you have tried, code samples used, etc.

Good luck!