woocommerce custom checkout field to add fee to order ajax

The post data is sent by the AJAX functions in 'post_data', serialized. So to get the value of your checkbox, you only need to parse_str() this!

parse_str( $_POST['post_data'], $post_data );

then you can get your 'add_gift_box' option from $post_data['add_gift_box']. Note that upon order completion, this 'post_data' element is not available anymore and everything is in $_POST.

Complete example, based on your code:

1) adding the checkbox to the checkout

add_action( 'woocommerce_after_checkout_billing_form', 'add_box_option_to_checkout' );
function add_box_option_to_checkout( $checkout ) {
    echo '<div id="message_fields">';
    woocommerce_form_field( 'add_gift_box', array(
        'type'          => 'checkbox',
        'class'         => array('add_gift_box form-row-wide'),

        'label'         => __('Ilość pudełek ozdobnych - 25 PLN/szt'),
        'placeholder'   => __(''),
        ), $checkout->get_value( 'add_gift_box' ));
        echo '</div>';
}

2) script to update cart when checkbox clicked (no need for extra AJAX requests!)

add_action( 'wp_footer', 'woocommerce_add_gift_box' );
function woocommerce_add_gift_box() {
    if (is_checkout()) {
    ?>
    <script type="text/javascript">
    jQuery( document ).ready(function( $ ) {
        $('#add_gift_box').click(function(){
            jQuery('body').trigger('update_checkout');
        });
    });
    </script>
    <?php
    }
}

3) action to add the fee

add_action( 'woocommerce_cart_calculate_fees', 'woo_add_cart_fee' );
function woo_add_cart_fee( $cart ){
        if ( ! $_POST || ( is_admin() && ! is_ajax() ) ) {
        return;
    }

    if ( isset( $_POST['post_data'] ) ) {
        parse_str( $_POST['post_data'], $post_data );
    } else {
        $post_data = $_POST; // fallback for final checkout (non-ajax)
    }

    if (isset($post_data['add_gift_box'])) {
        $extracost = 25; // not sure why you used intval($_POST['state']) ?
        WC()->cart->add_fee( 'Ozdobne pudełka:', $extracost );
    }

}

This is awesome!! Thanks a lot. I've changed it a little bit to add a percentage instead. I know this is not a better answer but I have no reputation to push your answer up. For whoever was stuck like me..

add_action( 'woocommerce_cart_calculate_fees', 'woo_add_cart_fee' ); function woo_add_cart_fee( $cart ){
  global $woocommerce;

    if ( ! $_POST || ( is_admin() && ! is_ajax() ) ) {
    return;
}

if ( isset( $_POST['post_data'] ) ) {
    parse_str( $_POST['post_data'], $post_data );
} else {
    $post_data = $_POST; // fallback for final checkout (non-ajax)
}

if (isset($post_data['add_gift_box'])) {

$percentage = 0.01;
$surcharge = ( $woocommerce->cart->cart_contents_total + $woocommerce->cart->shipping_total ) * $percentage;    
$woocommerce->cart->add_fee( 'Surcharge', $surcharge, true, '' );
}

}