Updating custom post meta with ajax

If you load a file directly none of the WordPress functions will work. That is why you should nearly always use the AJAX API. The AJAX API solves this problem. Everything loads in WordPress context.

You would wrap your processing PHP is a function:

function my_ajax_cb_wpse_108143() {
    $cote = get_post_meta($_POST['id'], 'cote', true);
    if($_POST['cote'] == 'like') {
        $newCote = $cote++;
        update_post_meta($_POST['id'], 'cote', $newCote);
    } else {
        $newCote = $cote--;
        update_post_meta($_POST['id'], 'cote', $newCote);
    }
}

Hook that into the AJAX system:

add_action('wp_ajax_my_update_pm', 'my_ajax_cb_wpse_108143');
add_action('wp_ajax_nopriv_my_update_pm', 'my_ajax_cb_wpse_108143');

Submit your request to http://site/wp-admin/admin-ajax.php and pass my_update_pm as an argument when the Javascript makes a request.

var data = {
    action: 'my_update_pm',
    id: jQuery('section#single article input:last-of-type').attr('value'), 
            cote: 'like'
};
jQuery.post(ajax_url, data, function(response) {
    // whatever you need to do; maybe nothing
});

You can set ajax_url similarly to this from the Codex:

add_action( 'admin_enqueue_scripts', 'my_enqueue' );
function my_enqueue($hook) {
    if( 'index.php' != $hook ) return;  // Only applies to dashboard panel
    wp_enqueue_script( 'ajax-script', plugins_url( '/js/my_query.js', __FILE__ ), array('jquery'));
    // in javascript, object properties are accessed as ajax_object.ajax_url, ajax_object.we_value
    wp_localize_script( 'ajax-script', 'ajax_object',
            array( 'ajax_url' => admin_url( 'admin-ajax.php' ), 'we_value' => $email_nonce ) );
}