The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the prerequisite knowledge of reading his two papers? I'm afraid basic Riemannian geometry would not suffice.
Here's a (possibly incomplete list) of some good things to know:
Riemannian geometry, including a good understanding of curvature.
Some algebraic topology, including familiarity with differential forms and De Rham cohomology. I first learned homology from Hatcher's Algebraic Topology and de Rham cohomology from Bott and Tu's Differential Forms in Algebraic Topology, but there are many other sources.
Vector bundles, and Chern-Weil theory of characteristic classes. Milnor and Stasheff's Characteristic Classes is a classic; there's an appendix on Chern-Weil theory, which may be enough to get you started.
Edti: The idea of characteristic classes is to study a vector bundle over a manifold $M$ by associating to the vector bundle various distinguished cohomology classes in the cohomology of $M$. Loosely, these classes are supposed to measure how "twisted" (or "nontrivial") the vector bundle is. One can take a purely topological approach to this subject, but Chern-Weil theory is a more geometric approach, in which the cohomology classes are represented by explicit differential forms that are constructed from the curvature of a connection on the vector bundle. I think one could make the following analogy: just as C-G-B connects the topology and geometry of a manifold, Chern-Weil theory connects the topology and geometry of vector bundles.
Perhaps I was wrong to include characteristic classes in the list above. Strictly speaking, you actually don't need to study characteristic classes (or even vector bundles) in general to understand the C-G-B theorem, but might be good to do so to gain a little perspective. Maybe you can start attacking the proof of C-G-B, and come back to them as necessary.
The $n$-form (for $n$-dimensional $M$) that appears as the integrand in the C-G-B theorem is precisely the differential form representing the Euler class of the tangent bundle $TM$ that one obtains via Chern-Weil theory. For surfaces, the form is $K \, dA$; for higher-dimensional $M$, the form is a more complicated curvature expression, but it is always obtained from a polynomial in the curvature called the Pfaffian.
Searching mathoverflow turned up a few relevant questions and answers (among others, there's this one, which has links to several good sources).
Finally (probably in the long term, after you've read some of the above), if you are interested in analysis and PDE, you may appreciate learning some index theory, which involves beautiful connections between analysis, geometry, and topology. The Atiyah-Singer index theorem is a broad generalization of Chern-Gauss-Bonnet; you can get a quick introduction on Wikipedia. I got an introduction to this field from John Roe's Elliptic Operators, Topology, and Asymptotic Methods.