How can I make my custom posts appear in their assigned category url?

The query_posts solution is not reccomended (see you don’t know query). You should use pre_get_posts hook instead. See:

function my_extend_category_by_custom_post_type( $query ) {
    if ( $query->is_category() && $query->is_main_query() ) {
        $query->set( 'post_type', 'any' );
    }
}
add_action( 'pre_get_posts', 'my_extend_category_by_custom_post_type' );

PS: this should go into your functions.php if (custom post type is set from there) or, preferably, from site specific plugin.