Your/Ext/etc/config.xml and Your/Ext/controllers/Checkout/CartController.php

CartController.php

<?php

// Your/Ext/controllers/Checkout/CartController.php
class Your_Ext_Checkout_CartController extends Mage_Checkout_CartController
{
    public function yourNewAction()
    {
    }
}

config.xml

<config>
    <!-- Your/Ext/etc/config.xml -->
    <frontend>
        <routers>
            <checkout>
                <args>
                    <modules>
                        <your_ext after="Mage_Checkout">Your_Ext_Checkout</your_ext>
                    </modules>
                </args>
            </checkout>
        </routers>
    </frontend>
</config>