First you make sure that the client libraries you use for your database are configured correctly - for some database, it's just a matter of installing the libraries, others might need to a mapping of symbolic names to ip numbers and ports.

Then make sure you can actually connect to the remote server using an interactive tool (most database have at least one interactive tool).

Then you change the connection string you use to create a database handle when using the DBI. Depending on the database server, this might require changes in hostname, port, databasename, username and/or password.