Standard answer: It depends.
If you have your own DNS it's great for your interneal network if you have lots of corporate websites internally. You can go directly to their internal address (if permitted). You can resolve things internally how you like, more flexibility etc. And for external addresses you just relay the request to the external provider.
In your case, you can set up the DNS server as the router itself in your DHCP commands and configure the router to resolve the names. This is typical in a home network with an ADSL router for example.
Take a look at this document:
http://www.cisco.com/en/US/tech/tk648/tk362/technologies_tech_note09186a00800c525f.shtml
HTH,
Ian
