If you have a Dynamic IP, or if a Static IP you need a 3rd Party DNS Updater software from a hosting site like Dyndns.org or No-IP.com.
Then you can host your own MySQL DB and use it.
Or you can for a FEE buy space on someone's mysql server.
If you choose to host your own DB, You just need to setup the Bind-Address=0.0.0.0 var in the my.ini(Windows) or my.cnf(Linux).
Then create a DB access user account and set that accounts privileges.