Sort appearence order in the_content()

You could try to add the Facebook-Like html code via this filter:

/**
 * Append Facebook Like to the content
 *
 * @param string $content
 * @return string $content
 */
function custom_the_content( $content )
{
    $fblike="<div class="fb-like" 
                    data-layout="button_count" 
                    data-action="like" 
                    data-show-faces="false" 
                    data-share="false">
              </div>";

    return $content . $fblike;
}

add_filter( 'the_content', 'custom_the_content', 9, 1 );

if the priority of the author box is 10.

Otherwise you could adjust the priority via the Biographia settings page:

so you would have:

priority("facebook like") < priority("author box")