How to check the first character in a string in unix

Many ways to do this. You could use wildcards in double brackets:
str="/some/directory/file"

if [[ $str == /* ]]; then echo 1; else echo 0; fi

You can use substring expansion:
if [[ ${str:0:1} == "/" ]] ; then echo 1; else echo 0; fi

Or a regex:
if [[ $str =~ ^/ ]]; then echo 1; else echo 0; fi