This requires you to override WooCommerce templates:
1) First read: Template structure & Overriding templates via a theme to understand how you can change some templates copying them to your active theme folder
2) Copy the template
templates/cart/cart.php located in your Woocommerce plugin to your active theme in
woocommerce/cart/cart.php (so not in a "template" subfoder).
3) Open edit the template
cart.php and make the necessary changes in the html structure, adding your custom column and related php code.