So far your approach is correct,
1) You will need to know the remote computer IP address or hostname
2) You will need to find out what services are running in the remote host.
3) You could use Nmap to scan for any open ports on the remote host.
4) Once you find what services and ports are opened.
5) Use Metasploit to exploit the host service.
6) Now to make the machine visible to the internet, there are two ways you can go about.
1) Open forwarding ports on the remote router.
2) Or place the remote host in a DMZ, this option is not recommended, because it will completely expose the host to the internet, however once this option is enabled, forwarding ports will no
longer be required.
If you want to go down this path, make sure the host does not have any confidential information or any important data on it.
Hope this helps.