Luckily it is easy to do, but you are slightly wrong here...
To begin with, these instructions depend on your definition of "Default".
By default, Remote Desktop is disabled and you must enable it from the remote tab in
System Properties (or click Windows Key + Pause/Break, then click
Once enabled, your machine will accept remote desktop connections on port 3389 (TCP) from anywhere. At this point, I would highly advise you set a static IP for your computer (unless your router allows NAT rules based on hostname/ddns).
In order to reach your computer from outside your LAN, all you have to do is set a NAT rule that forwards port 3389 (TCP) from the external IP to the internal one.
After you have done this, your machine will be accessible via your external/internet IP. Next, simply set up a DNS entry that points from computer-name.bearce.me to your internet IP.
The draw back to this is that you will need to either use a Remote Desktop Proxy (which is awkward to set up), or change the port manually as you can only forward the request from outside the network to one machine.
I am making some assumptions here - e.g. these are not directly bridged machines with external IPs. This does get quite advanced and I hope this points you in the right direction, but, happy to help with follow up questions.