Chern classes tangent bundle.

Question

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this example. I have a preliminar question: how can I imagine the first Chern classes $c_1 \in H^*(\mathbb{C}P^1, \mathbb{Z})$? I read that it is an hyperplane, but what hyperplane? Why is it the fundamental homology class? Now we have tha Chern classes are the coefficient of characteristic polynomial of the curvature form $(\Omega)$ of $T\mathbb{C}P^n$, where $\Omega:= d\omega + \frac{1}{2}[\omega,\omega]$ with $\omega$ the connection form. How can I calculate $\Omega$ for $\mathbb{C}P^n$ and why this coefficients are cohomology classes? And how can I calculate $c(T\mathbb{C}P^n)$? Futhermore, how can I imagine Chern classes as the obstruction to reduce the structure group of $T\mathbb{C}P^n$ $(U(n))$ to $SU(n)$? Can you explain me and explicit calculus?