How to set featured image as background for a specific category?

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I am not sure where you are trying to do this but give something like this a try.

<?php 
    $category_id = 45;
    query_posts('showposts=1&cat=".$category_id);
    if (have_posts()) : while (have_posts()) : the_post(); 
        $backgroundImg = wp_get_attachment_image_src( get_post_thumbnail_id($post->ID), "full' );
    endwhile; endif;
    wp_reset_query(); 
?>