However, your problem is probably more complex than that. If you want everyone on the Internet to be able to resolve the addresses in your DNS server, you'll need to be considered authoritative for the domains you manage and be part of the Internet DNS system. That topic is pretty complex, and I won't try to explain every step in this forum.
It's complicated enough that I personally do not manage my own DNS, even though I do have a domain. There are many free DNS providers that can have you up and running in a few minutes. I use www.zoneedit.com. Good luck with it.