This question is inspired by this question. It addressed the same problem, but the books recommend seemed to use a lot of tools I was not familiar with, so I'm having a lot of trouble following. Obviously, for a bounded $f\in \mathcal{L}^1([a,b])$ the function $$F:[a,b]\rightarrow \mathbb{R}, x \mapsto \int_{a}^x f(t)dt$$
is absolutely continuous. Now I want to show that absolute continuity does imply differentiability a.e. and $F'=f$ a.e.
From reading this I found that you could maybe show this result by showing that $F$ is of bounded variation, which implies it's a difference of monotonically increasing functions, which implies diff. a.e. . Is there a more elementary or direct approach to prove this result for example avoiding bounded variations and lebesgue points?
Well bounded variation is way a bigger class than AC. You can show by rather elementary means that any AC function on the line is the difference of two increasing function, and then prove that increasing functions are differentiable a.e.
The proof is very elementary and done by hands, see this for example: http://www.math.unipd.it/~monti/AR_2014_15/App_GDM_2013_14.pdf