The professor for one of my classes has no direction I'm in my final semester of a Computer Science degree at University and we have general education requirements. 

I'm in a Public Speaking course with an interesting professor, lets call him professor Johnson. Johnson doesn't seem to be actually teaching any material or guiding us on any topics that are related to the book or curriculum we're assigned. He was obviously given all of the online course material, including our online quizzes and study guides. Most of the quiz grades are handed to students, whether we completed them or not. Some are not even available.

He was a one year contract with the university, and was hired a mere two weeks prior to the semester beginning. He doesn't know how to use the learning management system and seems to have no interest in learning it. I inquired about an online quiz, and he had no idea what I was talking about or how to get to it. Considering I'm the only senior, I offered to stay after and show him around the site, suggesting that he attach due dates to quizzes so the students will actually do them on time with the reading and he can decide not to accept late assignments if he so chooses. He nodded and hasn't made any changes since.

We spend our days in class talking about no particular topics, nothing about the book, and no PowerPoints in sight. He rants about our news habits, how much water we like to drink, and we spend 25 minutes each period doing breathing exercises. The only related material is a monthly speech that we must give, with basically no guidance.

My question is whether I should speak to the department chair about his behavior. We're halfway through the course and the LMS hasn't seen a single bit of attention, he doesn't respond to emails of any topic, and asks the class what we want to discuss for the day. It's obviously very freestyle and he's not following any course material. I'm finding it harder and harder to attend class, and increasingly frustrating that I have spent money on this course. He seems very financially worried about his contract being renewed, as he has literally brought this up in class.

Should I leave an honest course review and leave it at that, let his own fate decide whether his contract is renewed, or speak with the department chair about my concerns?
Yes, you should raise this with the department, either with the Undergraduate Director or the Chair. The problem is not Prof. Johnson's unfamiliarity with the LMS--as a new faculty member, that is understandable. But you should restate what you said here about not teaching the assigned material. 

You mention online quizzes that were given to him. Are they being offered and graded? If so, is the grading fair? If not, what assignments will your course grade be based on? 

If the lessons are not being taught, students cannot complete the assignments successfully. That is the kind of issue the department's administrator will be concerned about, because they do want you to get a serious education in every class. And they know that the grading scheme in the syllabus, as well as the learning objectives, are contractual issues that have to be fulfilled. 

In talking to the administrator, stay focused on teaching performance issues only and avoid anything personal. You are supposed to be learning something critical about the history and practice of public speaking, and you should have an opportunity to demonstrate that learning fairly, through tests or papers. If that is not happening, the department's administrators need to know. The sooner the better.