Untitled

var data = {
    action: 'is_user_logged_in'
};

jQuery.post(ajaxurl, data, function(response) {
    if(response == 'yes') {
        // user is logged in, do your stuff here
    } else {
        // user is not logged in, show login form here
    }
});

function ajax_check_user_logged_in() {
    echo is_user_logged_in()?'yes':'no';
    die();
}
add_action('wp_ajax_is_user_logged_in', 'ajax_check_user_logged_in');
add_action('wp_ajax_nopriv_is_user_logged_in', 'ajax_check_user_logged_in');

<body <?php body_class(); ?>>
   //your html code
</body>

if ($('body').hasClass('logged-in')) {
       //execute your jquery code.
}

// Simplified... please note, that all names/vars/etc. in my class got unique names.
// ...the same goes for the script handler.
class wpse69814_example
{
    public $response;

    public function __construct()
    {
        add_action( 'wp_enqueue_scripts', array( $this, 'enqueue' ) );
        add_action( 'wp_enqueue_scripts', array( $this, 'localize' ), 20 );
    }

    public function enqueue()
    {
        wp_enqueue_script(
            'wpse69814_handler',
            plugins_url( 'url/to/file.js', __FILE__ ),
            array( 'jquery' ),
            filemtime( plugins_dir_path( __FILE__ ).'/path/to/file.js' ),
            true
        );
    }

    public function localize()
    {
        wp_localize_script( 'wpse69814_handler, 'wpse69814_object', array(
            'ajaxurl'    => admin_url( 'admin-ajax.php' ),
            'ajax_nonce' => wp_create_nonce( 'wpse69814_nonce' ),
            'action'     => 'wpse69814-handler-action',
            'data'       => array(
               'is_user_logged_in' => is_user_logged_in(),
            )
         )

    }
}

function login_function() {
    setcookie('wp_user_logged_in', 1, time() + 31556926, '/');
    $_COOKIE['wp_user_logged_in'] = 1;
}
add_action('wp_login', 'login_function');

function logout_function() {
    unset($_COOKIE['wp_user_logged_in']);
    setcookie('wp_user_logged_in', null, -1, '/');
}
add_action('wp_logout', 'logout_function');

if (document.cookie.indexOf('wp_user_logged_in') !== -1) {
    //do something when user logged in
} else {
    //do something when user logged out
}