The easiest way is to login into the Delphix GUI console, but on the login dialog first toggle from "Delphix Admin" to "Server Setup" and then login as "sysadmin".
On the main dashboard, you'll see a panel in the lower middle of the screen entitled "Network"...

Click on the "Modify" link in the upper-right corner of the panel to modify the "DNS Servers".
You can also login via SSH to the Delphix CLI into the "syslogin" account, then enter the command "/network setup get" to display the current settings, then enter "/network setup update" followed by "set dnsServers=
<list>" and "commit" where
<list> is a comma-separated list of the IP hostnames or addresses for the DNS servers.
Hope this helps?