I've gathered the gist of a particularly nice construction for Chern classes in a topological setting, but I can't quite figure out how to find the first class without making use of a classifying map. I've been told that this construction roughly goes through in AG without classifying maps (evidently first proposed by Grothendieck.)
Given a vector bundle $E \to B$ with fiber $V$, we form the projectivization $\mathbb P(E) \to B$, which has a tautological sub-bundle $L$, where fibers for $(x,\ell) \in P$ where $x \in B$ and $\ell \subset E_x$ is exactly $\ell$.
A formal descripton is given in $10.1.5$ here.
Now, supposing that we have a description of $\alpha \in H^2(P,\mathbb Z)$, and argue that powers of this element restrict to generators on $H^2(\mathbb CP^{n-1})$, and conclude with Leray Hirsch that $H^*(\mathbb P(E))$ is a free module over $H^*(B)$. Expressing $c_1(L)^n$ as a linear combination of the first $n-1$ powers gives the chern classes for $E$.
Question 1: How can one define $c_1(L) \in H^2(P,\mathbb Z)$ without using the classifying map $B \to \mathbb CP^{\infty}$ for line bundles?
Question 2: Can the following argument be made to work (of course by completing it?
Given the tautological bundle $L \to \mathbb P(E)$ one can use the association $Vect^1(\mathbb P(E)) \to \check{H^1}(\mathbb P(E))$ to obtain $\alpha \in H^1(P,\mathbb C^{\times})$. Is there a way to map from $H^1(\mathbb P(E),\mathbb C^{\times}) \to H^2(\mathbb P(E),\mathbf Z)$ and use this to get the chern classes?
The above argument can indeed be modified (and it appears that Grothendieck mentions this in his original paper.) This was also already mentioned by Lorenzo in the comments, but it took me a few days to (maybe) understand what was going on.
There is a short exact sequence of sheaves
$$A(X,\mathbb Z) \to A(X,\mathbb C) \to A(X,\mathbb C^{\times}) $$
that gives rise to a long exact sequence in sheaf cohomology where the connecting homomorphism $\delta:\check{H}^1(X,\mathbb C^{\times}) \to \check{H}^2(X,\mathbb Z)$, provides the isomorphism we needed.