Joining a professor's research group that is full
Yes, it is a good way to get to work under him. By taking the course, Prof. will get to know you. However, you should do well above average in his course to increase the chances of joining his research group. I'd suggest that around mid-point of semester you mention that you still want to join his group on Spring. There is no point on hurrying things up yet, specially because he already said that his group is full.