A class in Perl is nothing more than a namespace. All you need to do is add to that namespace:

use Data::Dumper;

# $d is an instance of Data::Dumper
$d = new Data::Dumper([]);

# Use a typeglob to create &Data::Dumper::foo
*{Data::Dumper::foo} = sub { print "foo\n" };

# And now $d has the method
$d->foo();

# Switch into the Data::Dumper package (namespace)
package Data::Dumper;
# Define a function here
sub bar { print "bar\n" }

# Back in main, $d has the new method
package main;
$d->bar();

# No different if the method exists
*{Data::Dumper::Dump} = sub { print "Not Dump\n" };
$d->Dump();
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Perlsub has an example of replacing the built-ins.