What you would need to do is setup your machine just like you want it. Then you have 2 options:
1. Purchase a domain name from a place like godaddy.com or another registrar. Then add your servers IP to the nameservers. (This only works if your server has a static IP address)
2. Use something like
http://www.dyndns.com/ to setup a domain name and then run a client on your server to auto update the IP address. (This works best for dynamic IP addresses)
As long as you can setup your LAMP server and run it already I don't see any problems with trying to make it public.
