By default, WordPress uses the _custom_background_cb() function to create the style you show in your question. You can use a function of your own by adding it to the add_theme_support() in your theme.
add_theme_support( 'custom-background', array( 'wp-head-callback' => 'wpse_113346_custom_background_cb' ) );
Initially, you can use the guts of the _custom_background_cb() function for your own function:
/**
* My custom background callback.
*/
function wpse_113346_custom_background_cb() {
// $background is the saved custom image, or the default image.
$background = set_url_scheme( get_background_image() );
// $color is the saved custom color.
// A default has to be specified in style.css. It will not be printed here.
$color = get_theme_mod( 'background_color' );
if ( ! $background && ! $color )
return;
$style = $color ? "background-color: #$color;" : '';
if ( $background ) {
$image = " background-image: url('$background');";
$repeat = get_theme_mod( 'background_repeat', 'repeat' );
if ( ! in_array( $repeat, array( 'no-repeat', 'repeat-x', 'repeat-y', 'repeat' ) ) )
$repeat="repeat";
$repeat = " background-repeat: $repeat;";
$position = get_theme_mod( 'background_position_x', 'left' );
if ( ! in_array( $position, array( 'center', 'right', 'left' ) ) )
$position = 'left';
$position = " background-position: top $position;";
$attachment = get_theme_mod( 'background_attachment', 'scroll' );
if ( ! in_array( $attachment, array( 'fixed', 'scroll' ) ) )
$attachment="scroll";
$attachment = " background-attachment: $attachment;";
$style .= $image . $repeat . $position . $attachment;
}
?>
<style type="text/css" id="custom-background-css">
body.custom-background { <?php echo trim( $style ); ?> }
</style>
<?php
}
At this point, you should test it to make sure it works okay. Once tested, you can start making changes to wpse_113346_custom_background_cb() to customize the output to your liking.