Putting the various suggestions together, the cleanest version I was able to come up with (without unreliable grep which triggers parts of words) is:
kill -0 $(pidof mysql) 2> /dev/null || echo "Mysql ain't runnin' message/actions"
kill -0 doesn't kill the process but checks if it exists and then returns true, if you don't have pidof on your system, store the pid when you launch the process:
$ mysql &
$ echo $! > pid_stored
then in the script:
kill -0 $(cat pid_stored) 2> /dev/null || echo "Mysql ain't runnin' message/actions"