I have NO experience with Mac OS, but I do have experience with DBI and DBD::mysql. The order in which things need to be installed is this:

  1. install MySQL database, *AND* make sure your can get
     it running and can interact with it via the "mysql"
     client.  On Red Hat Linux, the MySQL server package is
     separate from the MySQL client package - on the
     mysql server machine I usually install all the mysql
     packages there are.  For example, on my machine where
     I have MySQL up and running, I have these mysql packages
     installed:

        mysql-server-3.23.58-1.9
        mysql-devel-3.23.58-1.9
        mysql-3.23.58-1.9

     The 'mysql-3.23.58-1.9' package contains the MySQL client.

  2. make sure DBI is installed.
  3. install DBD::mysql
[download]

DBD::mysql requires that MySQL already be installed. The error you got

   Failed to determine directory of libmysqlclient.a|libmysqlclient.so
+. Use

  perl Makefile.PL "--libs=-L<dir> -lmysqlclient"
[download]

indicates that it can't locate the MySQL client ("libmysqlclient.so") library. So I believe you need to find the MySQL packages for Mac OS and install those, and get mysql working. Then move on to install DBI (if not already installed) and DBD::mysql.

HTH.