How to make current user’s avatar + display name link to their profile?

Use get_the_author_link() or get_the_author_meta('user_url', $user_id). Your code would become something like:

$author_link = get_the_author_link();
$header .= '<div><a href="'. $author_link . '">' .get_avatar($user_ID, 64).'</a></div>';
$header .= '<div><strong><a href="' . $author_link . '">' . __("", "front-end-pm").' '. fep_get_userdata($user_ID, 'display_name', 'id') .'</a></strong>';

Watch out with the quotes though, your code seems to use double quotes, I tried to replace it with single ones but you might have to check that in detail.

If this doesn’t work, use $author_link = get_the_author_meta( 'user_url', 64 ); on the first line. This depends if you’re inside the loop or not.