But that means I still have to retain my original code where I have it under:
private void btnCount_Click(object sender, EventArgs e)
is that right?
If you still want to have the same behavior on a button click that you do now, yes. But keeping a running count means you do not need to have a separate button that counts all vowels from scratch.
Btw, if I write the private int _vowels = 0; I got an error saying that expected token. What does it mean?
You may be adding it to the wrong place. _vowels is a field for your form class, not a local variable. The code I gave has it in the right place. Make sure that you do the same thing and it should work.