I can't talk for every Router out there but it should be standard that the router act as a relay. The PC will get the Gateway address of the router and the router will forward the request to the DNS-Server. That is how it should work with the majority of home routers. The only thing a router does is that it has a cache to store some of the DNS resolved addresses. So the PC if using DCHP doesn't know the IP-Address of the provider's server and the PC doesn't need to. The router does and will act as a relay.

That is how it works with routers from Dlink, Netgear, Sphairon, AVM etc.