You would first create the new site and make sure your content is available via that domain name. You may need to use a host file to test this. Your new domain name would use the same IP address as the old site. Apache and IIS, as well as others, examine the request packet to know which site a request is addressed for so this should work okay.
Here is where I differ from your thought.
I would redirect your old site to the new site immediately using a blanket 301 redirect. Your content will be available from both domains without having a duplicate content penalty from the search engines.
Here is what I warn you.
If there are any links to your old domain, the 301 redirect will retain the value of the link only as long as you maintain the old domain and the redirect. Otherwise, it will be lost. If there are valuable links to the old domain, make sure that you replicate the old link profile for the new domain where you can and seek new links. This way, whatever link value you had, you can retain at least some of it and add new link value prior to deleting the old domain.