Go with
www.zoneedit.com (they have free and paid services, depending on the usage patterns), they're really good, and they even support a whole range of dynamic IP clients.
The steps to get your domain resolving to your IP address are:
- Register the domain name.
- Set the nameservers in your WHOIS record to your DNS servers (be it ZoneEdit, your ISP's or your own). If you used an OpenSRS registrar (most small companies are nowadays), you can do it from https://manage.opensrs.net/. If you are unsure, ask your registrar.
- Edit your DNS server's zone settings accordingly (define A, CNAME, MX, e.t.c. records that suit your needs).
- Configure the web server to respond to HTTP requests from that domain name.
Note that once you update your WHOIS record, it may take anywhere from 24-96 hours to make the changes active, so allocate at least a week for this project just to be safe.
P.S. You need at least two unique DNS servers in your WHOIS. If you only have one box/one IP available, you have to go with your ISP's or ZoneEdit's DNS servers.