On Folland's Real Analysis book page $90$, the Lebesgue-Radon-Nikodym Theorem is given as
Let $\nu$ be a $\sigma$-finite signed measure and $\mu$ a $\sigma$-finite positive measure on $(X,\mathcal{M})$. There exists unique $\sigma$-finite signed measure $\lambda,\rho$ on $(X,\mathcal{M})$ such that $\lambda\perp \mu$, $\rho\ll\mu$, and $\nu=\lambda+\rho$. Moreover, there is an extended $\mu$-integrable function $f: X\to\mathbb{R}$ such that $d\rho=fd\mu$, and any two functions are equal $\mu$-a.e.
To prove this theorem, we first can consider the case that $\nu$ and $\mu$ are "finite" and "positive". Then, we can extend that to the case where $\nu$ and $\mu$ are "$\sigma-$finite" and "positive". Finally, since $\nu = \nu^+ - \nu^-$, we can conclude that for signed measure $\nu$.
But I have problem in understanding the second step. In this step, we can write $X = \cup_j A_j$ where $A_j$'s are disjoint and $\nu(A_j)< \infty$ and $\mu(A_j) < \infty$. Then, we can define, $\nu_j(E) = \nu(E \cap A_j)$ and $\mu_j(E) = \mu(E \cap A_j)$ where $\nu_j$ and $\mu_j$ are finite. So, from the results of the first step, we know that $\lambda_j\perp \mu_j$, $\rho_j\ll\mu_j$, and $\nu_j=\lambda_j+\rho_j$, $d\rho_j=f_jd\mu_j$. But then, it says that if we define $\lambda = \sum_j \lambda_j$ and $f = \sum_j f_j$, we have $\nu=\lambda+\rho$ where $d\rho = fd\mu$.
We know that $\rho_j\ll\mu_j$ and $\lambda_j\perp \mu_j$. To show that $\rho\ll\mu$ and $\lambda \perp \mu$, is it true to say that since for every $j$, $\rho_j\ll\mu_j$ and $\lambda_j\perp \mu_j$, then we can conclude that $\sum_j\rho_j\ll \sum_j\mu_j$ and $\sum_j\lambda_j\perp \sum_j\mu_j$?
The sets $A_j$ are disjoint, and so $\nu_j\perp\nu_k$ for $j\ne k$. Then by countable subadditivity $$\sum_j\nu_j(E)=\sum_j\nu(E\cap A_j).$$ Again, the sets $E\cap A_j$ are disjoint, so $$=\nu\left(E\cap(\cup_jA_j)\right)=\nu(E\cap X)=\nu(E).$$
Similar results apply to $\lambda_j$ and $\rho_j=f_jd\mu_j$. You can use this to show that $\lambda$, $\rho$ are mutually singular unsigned measures, and that $\lambda\perp \mu$, $\rho\ll \mu$. Just write them out as the sum, remembering that the $A_j$ are disjoint.
Does that help? The proof that you have written is complete, what in particular about it doesn't make sense to you?
EDIT: In response to your edits, we will show that $\rho\ll \mu$, which is equivalent to showing $\rho=fd\mu$. By definition of $\rho$ $$\rho(E)=\sum_j\rho_j(E).$$ As $\rho_j=f_jd\mu$: $$=\sum_j\int_X\chi_Ef_jd\mu.$$ As the $f_j$ are positive we can use the Monotone Convergence Theorem to bring the sum inside the integral: $$=\int_X\chi_E\sum_jf_jd\mu=\int_X\chi_Efd\mu=\int_Efd\mu.$$ Thus $\rho=fd\mu$.
To show $\lambda\perp\mu$, suppose $E$ has $\mu$-measure zero, and define $E_i=E\cap A_i$. The result them follows from writing out $\mu=\sum_i\mu_i$, and noting that $\mu_i\perp\mu_j$ for $i\ne j$.