This can be solved simply by viewing the set as a circular list. Just pick one of the next N-1 characters.

# No two repeated characters

my @set = qw( a b c d e f );
my $len = 12;

my $s = '';
my $i = int(rand(@set));
for (1..$len) {
   $s .= $set[$i];
   $i = ( $i + int(rand($#set)) + 1 ) % @set;
}

print("$s\n");
[download]

dbadfbabfbcd
deabfadaeafb
edbaceabacad
[download]

Update: Fixed bugs, then I realized I misread the question. On the plus side, the solution can easily be adapted to solve the actual problem.

# No more than two repeated characters

my @set = qw( a b c d e f );
my $len = 12;

my $s = '';
my $i;
for (1..$len) {
   if ($s =~ /(.)\1\z/) {
      $i = ( $i + int(rand($#set)) + 1 ) % @set;
   } else {
      $i = int(rand(@set));
   }
   $s .= $set[$i];
}

print("$s\n");
[download]

aadbadfafbfb
dbfaefababea
efeabbcebbee
[download]