You need to load the current user before you attempt to render the field:

<?php
  global $user;
  print render(field_view_field('user', $user, 'field_avatar'));
?>

You need to reference the $author of the comment rather than the currently logged in $user. Luckily, $author comes as a parameter to the tpl file, try this?:

<?php
  print render(field_view_field('user', $author, 'field_avatar'));
?>