Active directory cannot function without setting up a DNS server. During the setup of AD, a DNS server was configured on that machine.
You cannot use the DNS server your ISP uses because it has no reference to your computer names.
First, verify that DNS is up and running on the Domain controller you setup. Next, on your remote machine, manually input the IP address of your controller. After doing so, you should be able to ping any computer including the server by name. Second, you should create a forwarding reference inside of your DNS server to point to the IP address of your ISP DNS. This allows all unresolved name requests to be forward to your ISP because any unresolved requests should be for outside internet traffic.
Hope this helps and points you in the right direction