A few things:
1. Do not use the DHCP from the Router on client PCs as they will give the wrong DNS addresses.
2. Configure clients using a static IP address and configure DNS to point to DNS on your DC Server (if you still want DHCP, enable and configure DHCP on the DC server, and disable it on the router).
3. If you have pointed client to DC's DNS server but cannot get internet access, configure Forward address on DNS (see properties of server) to point to router's IP address or usual ISP's DNS addresses.