For those who have the same problem and want to show badges under the name of the achievement, I figured it out:
Put this line of code into the loop-single-achievement.php file located under wp-content/plugins/achievements/templates/achievements/achievements. You should see this on line 21:
<a href="<?php dpa_achievement_permalink(); ?>"><?php dpa_achievement_title(); ?></a>
Underneath that, put this:
if ( has_post_thumbnail( $notification->ID ) )
echo '<a href="' . esc_url( $url ) . '">' . get_the_post_thumbnail( $notification->ID, 'medium', array( 'class' => 'attachment-medium dpa-achievement-unlocked-thumbnail' ) ) . '</a>';
That puts the featured image from that achievement underneath. Hope this helps somebody!