What can I do as a graduate student to maximize my chances of obtaining a professorship?
I don't think there's anything "special" that's unique to applying to academia that a graduate student can do to increase the odds of becoming a professor in the long run, particularly if one is going to do a postdoc later on.

The two areas that might help are: 


Gain teaching experience that goes beyond the standard "recitation section" leader—that is, into actual lecturing and other forms of direct interaction with students, as well as formulation of assignments and examinations. This might make a difference at schools which are more teaching-oriented rather than research-oriented.
Formulate a well-defined scope for your future research activities, and also develop the tentative outlines for the first few projects that you'd start in that field. This is an essential part of any professorial application, and the sooner you start working on it, the more polished it will be when you're ready to apply for positions when the time comes.


Beyond that, what makes someone a good candidate for a postdoc are essentially the same qualities that will help in being a good candidate for a professorship later on.