When you mention API testing, are you implying Class::DBI? All of my classes are descended from Class::DBI - I've added no functionality to it.
API testing means that you should test how your class might be used. If a user calls the view_table method, it will spit out the data @blah. Even if Class::DBI does all the dirty work, you still want to be sure that your return values comes out as you expect. In a way this is black box testing since you are not examing the internals of Class::DBI (where the work is being done), but only checking that given the proper input, your methods return the proper output.
And then of course, you can use DBD::Mock to check that the right SQL statement was produced, and all that good stuff too. In a way this is also testing that Class::DBI is doing it's job correctly, although maybe more grey box testing.
