Different approaches to Jordan Canonical Form

Question

I know two different proofs of the existence of JCF. Let $V$ be a finite-dimensional vector space over base field $\mathbb{C}$ and $\alpha \in \mathsf{End}_{\mathbb{C}}(V)$.

1. Given transformation $\alpha$ we can define the action of polynomial $f(z) \in \mathbb{C}[z]$ on $v \in V$ as $f(z)v := f(\alpha)(v)$. This turns $V$ into a $\mathbb{C}[z]$-module.
2. Since $\mathbb{C}[z]$ is a PID, $V \cong \oplus_{i,j} \frac{\mathbb{C}[z]}{(p_i(z)^{r_{ij}})}$, where $p_i(z) \in \mathbb{C}[z]$ are monic and irreducible and $r_{ij} > 0$ (polynomials $p_i(z)^{r_{ij}}$ are called 'elementary divisors').

Question 1. Why after going from $V$ to $\oplus_{i,j} \frac{\mathbb{C}[z]}{(p_i(z)^{r_{ij}})}$ via this isomorphism, operator $\alpha \colon V \to V$ still acts via multiplication by $t$?

UPD (Answer to Q1). If $\pi\colon V \to \oplus_{i,j} \mathbb{C}[z] / (\cdots)$ is iso, then $$(\pi \circ \alpha \circ \pi^{-1})(v) = \pi(\alpha(\pi^{-1}(v))) = \pi(t \pi^{-1}(v)) = t \pi(\pi^{-1}(v)) = t v$$ by definition of $\alpha$ and linearity of $\pi$. (Sorry, it was obvious).

3. Hence it is easy to see that characteristic polynomial $\chi_\alpha(z)$ is equal to $\prod_{i,j} p_i(z)^{r_{ij}}$, and $p_i(z) = z - \lambda_i$. Moreover, $\alpha$ acts on each summand $U := \frac{\mathbb{C}[z]}{(z - \lambda)^r}$ with multiplication by Jordan cell $J_{\lambda, r}$ if we choose $$(z - \lambda)^{r-1}, (z-\lambda)^{r-2}, \ldots, (z - \lambda)^0 = 1$$ for basis of $U$.

The second approach is as follows.

1. We can prove that $V$ is the direct sum of its root subspaces $V = \oplus_{i = 1}^s V^{\lambda_i}(\alpha)$.
2. We can prove that for each nilpotent operator $\beta := \alpha - \lambda \cdot I$ the space $V^\lambda(\alpha)$ is the direct sum of some $\beta$-cyclic subspaces and on each of them $\alpha$ acts with multiplication by some Jordan cell $J_{\lambda,-}$.

If the first approach was chosen, then I don't understand how to obtain Jordan basis in terms of initial basis of $V$; using second approach it is an easy process, but the first approach seems more natural to me.

Question 2. How could we construct the Jordan basis of $V$ or, more specifically, how could we construct the isomorphism between $U := \frac{\mathbb{C}[z]}{((z - \lambda_i)^{r_{ij}})}$ and $r_{ij}$-dimensional cyclic subspace of $V_{\lambda_i}(\alpha)$? Is the corresponding subspace actually $\pi^{-1}(U)$?

Thank you in advance for any help!