This solution inverts the hash like this:

 k1   v1 k2   v2 k3   v1     v1           v2
(a => 1, b => 2, c => 1) => (1  => [a,c], 2 => [b])
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Then it just becomes a matter of finding those value-keys who point to more than a single original key name.

use strict;
use warnings;

my %hash = (a => 1, b => 2, c => 1, d => 3, e => 5);

my @non_unique = duplicated_values_keys(%hash);

print "$_\n" for @non_unique;

sub duplicated_values_keys {
    my %hash = @_;
    my %inverted;
    push @{$inverted{$hash{$_}}}, $_ for keys %hash;
    return map {@{$inverted{$_}} > 1 ? @{$inverted{$_}} : ()} keys %in
+verted;
}
[download]

Outputs:

a
c
[download]

...in no particular order.