One optimization right off the bat, if the string contains much more than 52 characters:

s/([a-zA-Z])/{ $count{$1}++; $1 }/eg while <>;
foreach (A .. Z) {
    $count{lc($_)} += $count{$_} || 0;
}
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Disclaimer: untested, but theoretically valid. (corrected on 15 April as turnstep noticed a typo)