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