adding tax_query to $query_string

the best way of doing this is WP_Query

$query = new WP_Query(
'post_type' => 'post',
'tax_query' => array(
        array(
            'taxonomy' => 'my_custom_tag',
            'slug' => 'slug',
            'terms' => array( 'my_tag_slug' ),
            'operator' => 'in',
        ),
    )
);

and you should do the ajax callback like this

add_action( 'wp_ajax_my_action', 'my_action' );

function my_action(){
//my custom code ...
}