use strict;
use warnings;

my $file1 =
"Wilder    1933    Col2    Col3    xyz
Pitney    1940    Col2    Col3    xyz
Hackman    1930    Col2    Col3    xyz
Kelly    1912    Col2    Col3    xyz";

my $file2 =
"Hackman    Col2    Col3    xyz
Kelly    Col2    Col3    xyz
Simmons    Col2    Col3    zyx
Wilder    Col2    Col3    xyz";

open my $fh1, '<', \$file1;

my %gene_dob;
while ( my $line = <$fh1> ) {
  my ( $gene, $dob ) = ( split /\t/, $line )[0, 1];
  $gene_dob{ $gene } = $dob;
}

close $fh1;
open my $fh2, '<', \$file2;

while ( my $line = <$fh2> ) {
  chomp $line;
  my $gene = ( split /\t/, $line )[0];
  $line .= "\t$gene_dob{$gene}" if $gene_dob{ $gene };
  print "$line\n";
}
[download]

Note that this code omits to issue any warning if any of your Genes are present in file1 but not in file2, or vice versa. Error checking is left as an exercise...

Update: Yeah, well. Marshall beat me to it by a couple of hours for the algorithm. My excuse: I was researching birth dates ;).