How can I change the fields displayed by dataTables?
If you look in CRM/Core/xml/Menu/Contact.xml you'll see that the civicrm/ajax/contactrelationships maps to CRM_Contact_Page_AJAX::getContactRelationships, so you can go into CRM/Contact/Page/AJAX.php and change what the getContactRelationships function does.

If you're not familiar with the techniques here to override core files, as opposed to editing the actual core files, you also want to take a look at that.