I may be missing the point of your question (seems a little broad).
But, first you need to own or control the domain name itself. Your
domain registrar usually provides a DNS service that then allows you
to point your domain to your servers IP address. That is where
Liferay would live.
So, is your question about how to find the liferay site with a
domain name? Is the question that you want the site to look like what
you posted above (Blog maybe?) Or do you want to use the
domain.subdomain.subdomain syntax shown?
Not exactly sure what you are looking for
