Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?

For a complete proof of the Gauss-Bonnet-Chern for arbitrary vector bundles (not just tangent bundles) see Section 8.3.2 of these notes. The proof is Chern's original proof, based on Chern-Weil theory, but the language is more modern.

For a purely topological proof, see Section 5.3 of these notes.

One reference that seems fairly good and that I just found by googling those key words is https://web.archive.org/web/20100524152105/http://www.math.upenn.edu/~alina/GaussBonnetFormula.pdf.

The first time I learnt this, however, was with these lecture notes: F. Mercuri, P. Piccione, D. V. Tausk, Notes on Morse theory, Publicações Matemáticas do IMPA, Rio de Janeiro, 2001, ISBN 85-244-0178-8; which maybe a little hard to find, but are very nicely written and I like them very much. Though, the proof you are looking for should be widely available elsewhere (google gives thousands of results, and I only looked at the first ones)...