There is really not a whole lot that is different. The DNS needs to be authoritative for the zone. In most cases it is likely that you will only be able and interested in forward domain look ups. If you want reverse look ups to resolve to your domain you will need to talk to you ISP. One thing to consider with the public DNS is that RFC 1918 addresses, such as those used on a LAN won't route and should not be returned in response to a query. Instead only the public IP addresses should be given out. You will need to have port 53 open to the public for the DNS to be available. Once you can connect to it on your public interface and get resolution, you can then go to your domain registrar and update your registration to indicate that your server is the DNS for this domain, which will then propagate through the DNS system, typically in about 24 hours. At this point, you are using your own public DNS.
If you decide to do this, you should also research how to properly secure it, for example understand how to run it in a chroot environment.