I have been reading through the wikipedia article about Chern classes and it currently has a section devoted to the Alexander Grothendieck axiomatic approach. The language used throughout the section ("He shows using the Leray-Hirsch theorem (...)") seems to imply that there is a paper or a book written by Grothendieck himself containing the results mentioned.
Thus, my question is: does such a paper exist? Did Grothendieck ever write about the topological Chern classes?
I started my search by reading through a paper referenced in the wikipedia article, La théorie des classes de Chern and it very well may be what I am looking for. I can't tell, because, being written in french, it is basically unreadable to me. Still, I have a feeling that this is not the right paper, because it doesn't reference the aforementioned Leray-Hirsch theorem and the notation used throughout highly suggests that this is the algebro-geometrical case.
That is the right paper! The "problem" is that he doesn't explicitly use Leray Hirsch, because he is constructing Chern classes for a suitable category of non-singular varieties over any algebraically closed field. In the paper, he essentially asserts Leray-Hirsch as Axiom A1 because that's really all he needs, but if you replace varieties over $k$ with complex manifolds and the chow ring $A(X)$ with the cohomology ring $\bigoplus_{i}H^{2i}(X,\mathbb{Z})$ you should be able to prove Axiom A1 using Leray-Hirsch, and that's the main use of the theorem in his approach to Chern classes. Chapter 4 of Bott and Tu follows Grothendieck's approach quite closely and I believe they do this proof.