Though it's worth knowing that unlike in scheme, common lisp will return the value after a setf. There's also a convenience macro called incf that increments variables so you can write the whole thing like this:
(let ((c 0))
(defun my-counter! ()
(incf c)))
And your other question: Why the different placing of the let?
In common lisp, defun, defvar, defparameter, defmacro, ... all affect global scope, no matter where they appear. scheme's define does not affect global scope; its effects are only visible locally. This means that a defun inside of a let body still creates a globally accessible function that closes over the variables defined in the let bindings. Scheme, by contrast, needs to have a define at global level (or at least outside the let) but the function body still needs to close over the let variables.
In Common Lisp, as opposed to Scheme, it is not possible that the car of the compound form to be evaluated is an arbitrary form. If it is not a symbol, it must be a lambda expression, which looks like (lambda lambda-list form*).