Add a menu item to the logged in users author page?

You can use wp_nav_menu_items filter for this, and check if user logged in, and use get_author_posts_url($autho_id) as bravokeyl suggested:

add_filter('wp_nav_menu_items', function( $items, $args ) {
    $index = 'top-menu'; // menu index key, if you're not sure then var_dump( $args->theme_location )
    global $current_user;
    if ( $current_user->ID && $index === $args->theme_location ) {
        $items .= sprintf(
            '<li id="menu-item-my-posts" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-my-posts menu-item-my-posts"><a href="https://wordpress.stackexchange.com/questions/247427/%s">%s</a></li>',
            get_author_posts_url( $current_user->ID ),
            'My Posts'
        );
    }
    return $items;
}, 10, 2);

Hope that helps.