which of the following is correct to use?

Let's see exactly how this process is working.

Here exactly modifies see, and the sentence stresses the manner in which you see the working of the process. (Case of an Adverb modifying a Verb).

Let's see how this process is working exactly.

Here exactly modifies how this process is working, and the stress shifts from see to how this is working. In other words, you are interested in knowing the manner in which the process is working. (Case of an Adverb modifying an Adverb Phrase).

The second version above could also be rephrased as—

Let's see how exactly the process is working.

Here exactly modifies how (Case of an Adverb modifying another Adverb).