First, we need to get the integer value for the character.
char='b' charint=$(printf "%d" "'$char'")
Then, we increase the integer by one
Last, we can get the new character by converting the integer to the character
newchar=$(printf "\x$(printf %x $charint)")
Here, we use the inner
printf to convert the integer to hexadecimal value and the outer
printf to print the character from its hexadecimal value.
The result is as follows.
$ echo $newchar c