Let’s add a new field to checkout, after the order notes, by hooking into the following.

/**
 * Add the field to the checkout
 */
add_action( 'woocommerce_after_order_notes', 'my_custom_checkout_field' );

function my_custom_checkout_field( $checkout ) {

    echo '<div id="my_custom_checkout_field"><h2>' . __('My Field') . '</h2>';

    woocommerce_form_field( 'my_field_name', array(
        'type'          => 'text',
        'class'         => array('my-field-class form-row-wide'),
        'label'         => __('Fill in this field'),
        'placeholder'   => __('Enter something'),
        ), $checkout->get_value( 'my_field_name' ));

    echo '</div>';

}

Next we need to validate the field when the checkout form is posted. For this example the field is required and not optional.

/**
 * Process the checkout
 */
add_action('woocommerce_checkout_process', 'my_custom_checkout_field_process');

function my_custom_checkout_field_process() {
    // Check if set, if its not set add an error.
    if ( ! $_POST['my_field_name'] )
        wc_add_notice( __( 'Please enter something into this new shiny field.' ), 'error' );
}

Finally, let’s save the new field to order custom fields using the following code.

/**
 * Update the order meta with field value
 */
add_action( 'woocommerce_checkout_update_order_meta', 'my_custom_checkout_field_update_order_meta' );

function my_custom_checkout_field_update_order_meta( $order_id ) {
    if ( ! empty( $_POST['my_field_name'] ) ) {
        update_post_meta( $order_id, 'My Field', sanitize_text_field( $_POST['my_field_name'] ) );
    }
}

The field is now saved to the order.

If you wish to display the custom field value on the admin order edition page, you can add this code.

/**
 * Display field value on the order edit page
 */
add_action( 'woocommerce_admin_order_data_after_billing_address', 'my_custom_checkout_field_display_admin_order_meta', 10, 1 );

function my_custom_checkout_field_display_admin_order_meta($order){
    echo '<p><strong>'.__('My Field').':</strong> ' . get_post_meta( $order->id, 'My Field', true ) . '</p>';
}