Usually customers who choose to work against an external database do so so that their database administrators can regularly manage the database and connect to it with management tools.
In order to connect into remote MySQL, you may need to change the 'bind-address' parameter in the MySQL my.cnf file and replace the GRANT ALL on artdb.* TO 'artifactory'@'localhost' IDENTIFIED BY 'password'; with the real Artifactory machine host or with any host (using '%').
If you have a DBA, we recommend that you let them create the schema and the user, and make sure to allow the remote client user (Artifactory) to be able to connect to the DB.
See also: Changing the Default Storage