Get logged in username in wordpress url

I think this will do what you are looking for.

add_action("template_redirect", 'template_redirect');
function template_redirect() {
 global $wp;    
     if(is_user_logged_in() && !isset($_GET['user'])){
    $current_user=  get_userdata(get_current_user_id());
    wp_redirect(add_query_arg(array('user'=>$current_user->user_login),home_url($wp->request)));
    die();
}
}