By the way, now that you have the full-featured, maximally advanced, subtle and idiomatic solution for using variables with "tr///", here's a simple, direct answer to your opening question:

eval "\$var =~ tr/$pattern1/$pattern2/";
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Note the use of double quotes around the expression, as well as the backslash in front of "$var". The double quotes will cause interpolation of variables (variables in the string will be replaced by their values); the backslash causes the "$" in "$var" to be passed as a literal, so "eval" gets a string that begins with the four characters '$var', rather than one that begins with the current value (contents) of the variable "$var".