H1 tags for header image?

try is_front_page() conditional function

<?php } else { ?>
         <hgroup>
              <?php $is_front_page = is_front_page() ? 'h1' : 'h6'; ?>
              <<?php echo $is_front_page ?> class="site-title"><a href="https://wordpress.stackexchange.com/questions/88142/<?php echo home_url("https://wordpress.stackexchange.com/" ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="home"><?php bloginfo( 'name' ); ?></a></h1>
              <h2 class="site-description"><?php bloginfo( 'description' ); ?></h2>
         </hgroup>

   <?php } ?>

Edit

To address the OP’s question about adding to the header image:

<?php $header_image = get_header_image();
if ( ! empty( $header_image ) ) { ?>
    <a href="https://wordpress.stackexchange.com/questions/88142/<?php echo esc_url( home_url("https://wordpress.stackexchange.com/" ) ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="home">
        <img src="<?php header_image(); ?>" width="<?php echo get_custom_header()->width; ?>" height="<?php echo get_custom_header()->height; ?>" alt="" />
   </a>
   <?php 
   /* Start new code here */
   $heading = ( is_front_page() ? 'h1' : 'h6' );
   ?>
   <<?php echo $heading; ?>>INSERT TEXT HERE</<?php echo $heading; ?>>
   <?php 
   /* End new code here */ 
   ?>
<?php } else { ?>