How can I do a url redirect to include a wordpress username?

You have to add action to, for example, template_redirect action hook. The action would perform your desired checks and redirect user using the wp_redirect() function. The code could look something like this:

function my_redirect_function() {
    // Check if home page is being displayed
    if ( is_home() ) {
        global $userdata;
        get_currentuserinfo();
        $username = $userdata->user_login;
        $url="http://my.url.org/mentions/" . $username;

        wp_redirect( $url );
        exit;
    }
}
add_action( 'template_redirect', 'my_redirect_function' );

You place whatever checks (such as conditional tags) you’d like instead of is_home() check.