To have your clients resolve names via your DNS server (which you want to do so that they will be able to resolve the domain), you need to allow your DNS server to forward queries or do recursive name resolution. There are a few ways you can do this.
By far the easiest way is to specifically allow the DNS server to make DNS requests (on port 53) to the internet, and either set up forwarders (eg. to your ISP), or configure it to do its own recursion. You don't need to open any other connectivity for this, and there is no appreciable security risk from it. Then, configure your DC as the DNS server for the workstations.
You can also set up another server to do DNS. Configure it to forward requests for names in your domain to your DC, and forward the rest. Configure this server as the DNS server for your workstations and your DC. Doing this complicates your network and provides no material benefit over the previous solution.
Third, you could set up a DNS proxy on your LAN which the DC can use as a forwarder. The DNS proxy is basically a DNS server which forwards its requests to your ISP's DNS servers. Of course, this server could also do its own recursive queries. Set this server as the forwarder on your DC, and have clients use the DC as a DNS server. This also is unnecessarily complex, but will work.
Keep in mind that having only one DC and having that DC use itself as its only DNS resolver is definitely not good practice and tends to create reliability issues. You should really have two of them, and have them resolve DNS for each other, with themselves as a backup.
Using residential gateway style equipment provided by ISPs (the modem/NAT-router/WAP/switch/firewall combo people erroneously call a router) is also usually bad practice and will tend to create reliability issues.