ts10:
I will leave everything outside of what you are asking out - but ensure that your server has been secured before you open it up to the world.
You will have to check with the company that you registered your domain name with - for instance Network Solutions.
With NS they have 'Custom DNS' solutions, as do most of the major domain registrar's, they will allow you to configure your domain name to point to your server's public IP address.
Please note: You must have a static IP address for this to work, if you are planning on hosting this off of DSL or Cable Modem, please be sure to check with your provider that you are allowed to host websites, as well that you have a static IP address.
Your computer CAN host DNS for your domain, although I don't recommend yet, it is usually recommended that you have 2 separate Name Server's to handle all requests, as you may want to have your e-mail or mailserver reside on another box/server.
In short:
Your first step would be to contact the company that sold you the domain name, as they will most likely provide DNS Services.
Once you have found a viable DNS Service, be sure that you have a static IP address that you can dedicate to web server.
A Home based router will allow you to share the IP address but isnt designed to handle any appreciable amount of traffic, and it is not recommended.
And finally, once you know / have the IP address you wish to use, you can provide it to your DNS Company (or registrar) they usually have a control panel page to handle these requests.
You can often wait UP TO 72 hours for the changes to take effect so be patient.
If you do not have a static IP address, do a google search for
Dynamic IP DNS Services" - there are many companies out there which will allow you to have a small client on your server to ensure that the DNS is always pointing to your server.
I hope I was of some help - and if you have further questions, don't hesitate to ask.