I don't have that book, but as far as I can understand, the “circularity” must mean this: in the phrase “iterating the successor operation a finite number of times”, we should mean a number of times corresponding to a natural number. But since the natural numbers are what we are defining, this is circular. So one has to define the natural numbers without reference to the concept of “finite”. Where the circularity is broken is if you rewrite your definition as follows:

1. 0 is a natural number,
2. the successor of any natural number is a natural number, and
3. nothing is a natural number unless it must be, by 1 and 2.

(All this stated in more technical language, of course.)

There is no reference to the notion of “finite” here. Instead, number 3 above gives us, by definition, the principle of induction. For example, how do you show that some object X is _not_ a natural number? Well, if for some property P, you can show P(0) and you can also show that ∀n: P(n)⇒P(n') where the prime denotes the successor function, but X fails property P, then you can know for sure that X is not a natural number.