Is the URL a domain you own, or is it a domain assigned to you by this service?
The free services usually let you pick your own sub-domain name to attach to one of the domain names that they own. For example, here is a list of domain names available through DynDns.com:
http://www.dyndns.com/services/dns/dyndns/domains.html
You would need to check if your desired sub-domain name is not all ready in use for a given domain name. For example, you could check if surfpipe.gotdns.com was available and claim it for yourself if available. Note that you would not legally have any ownership of the"surfpipe.gotdns.com" domain name.
Would you mind giving me the basic steps on how to forward the port through the router?
Have the camera configured to have a static IP address on your LAN or have the router reserve a specific DHCP LAN address for the camera based on its MAC address.
Check the camera documentation to find out which ports it uses for viewing and control and if the ports use TCP or UDP. The camera may also allow you to change its default port assignments.
Configure the router to forward given port numbers on the WAN side to the camera ports on the LAN side. Port numbers on the WAN side do not need to match the camera port numbers on the LAN side.
If your broadband modem is running NAT, you will also need to configure port forwarding on it as well. If the modem is running as a bridge, you should not need to worry about port forwarding on the modem.
If your router supports Dynamic DNS services, you can use it keep your public IP address up to date with the service. Otherwise you will need to keep a computer on your LAN turned on and running a Dynamic DNS updater program.