I would follow Chern in his short paper Vector bundles with a connection, where he introduces characteristic classes as polynomials in the curvature of a connection, and then shows (easily) their basic properties, and applies them to prove Gauss-Bonnet on surfaces. A student who has seen principal bundles and connections can grasp this in under an hour. (The main problem is to get to an application before too long, so that students are not confused as to the purpose of these strange expressions.) Then you can demonstrate the general properties of the classes. In a subsequent lecture, show that vector bundles are pulled back from Grassmannians, so that you can explain how to define integer characteristic classes, and show that over complex number coefficients they agree with the Chern classes as defined by Chern. Finally, you might discuss classifying spaces, depending on the students' backgrounds.