add $current_user to the end of URL

It depends on what is creating the menu bar. Most themes use wp_nav_menu() to display menus. If this is you, the filter ‘wp_nav_menu’ can be used to add additional HTML to the menu just before it is echoed out. The function has several other actions and filters that give you even greater control. Your best bet for those is to examine the source code.

You don’t mean the admin bar do you? There’s a completely different set of hooks for that.