Although it might not be the most efficient solution, this sounds like a job for recursion. First get all the top_level categories. Then loop through each one getting the children of that category.

# this assumes that you have the top level categories with a
# parent_id that is NULL
my $sql = qq~SELECT category_id, parent_id, category_name 
    FROM ss_catalog_categories~ WHERE parent_id IS NULL;
my $sth = $dbh->prepare($sql);
$sth->execute;
my $results = $sth->fetchall_arrayref({});
my $categories = format_results( $results );

sub format_results {
  my $results = shift;
  foreach my $result (@$results) {
    my $children = get_childrent($result->{category_id});
    my %category = (
      name     => $result->{category_name},
      children => $children,
    );
    push(@$categories, \%category); 
  }
}    

sub get_children {
  my $parent = shift;
  my $sql = qq~SELECT category_id, parent_id, category_name 
    FROM ss_catalog_categories~ WHERE parent_id IS ?;
  my $sth = $dbh->prepare($sql);
  $sth->execute($parent);
  return format_results($sth->fetchall_arrayref({}));
}
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This should at least give you an idea to go on - remember, this code is untested, but I've used the same idea before lots of times.