How to replace one character with two characters using tr

No, tr is specifically intended to replace single characters by single characters (or, depending on command-line options, to delete characters or replace runs of a single character by one occurrence.).

sed is probably the best tool for this particular job:

$ echo "asdlksad ~ adlkajsd ~ 12345" | sed 's/~/~\n/g'
asdlksad ~
 adlkajsd ~
 12345

(Note that this requires sed to interpret the backlash-n \n sequence as a newline character. GNU sed does this, but POSIX doesn’t specify it except within a regular expression, and there are definitely older versions of sed that don’t.)