How can I count the occurrences of a string within a file?

The number of string occurrences (not lines) can be obtained using grep with -o option and wc (word count):

$ echo "echo 1234 echo" | grep -o echo
echo
echo
$ echo "echo 1234 echo" | grep -o echo | wc -l
2

So the full solution for your problem would look like this:

$ grep -o "echo" FILE | wc -l