Solution:
A few template files make up the checkout page and can be found in:
wp-content/plugins/woocommerce/templates/checkout
The files which may be of most interest to you could be either form-checkout.php or review order.php.
Inside your own theme folder, create a folder called woocommerce and copy across any files you want to edit, making sure to place them inside the exact folder structure they are kept within Woocommerce. So in this case, copy the files to your_theme/woocommerce/templates/checkout/. Any changes made to these files will override the original template.
You can edit the templates and add, or remove, any code as you need to.