Definition: if $M$ is a smooth manifold, define the orientable double cover of $M$ by:
$$\widetilde{M}:=\{(p, o_p)\mid p\in M, o_p\in\{\text{orientations on }T_pM\}\}$$
together with the function $\pi:\widetilde{M}\to M$ with $\pi((p,o_p))=p$.
There are three things I'm trying to understand about $\widetilde{M}$:
What is its differentiable structure?
Why is $\widetilde{M}$ orientable?
Why is the connectedness of $\widetilde{M}$ equivalent to the non-orientability of $M$?
Here's where I'm at: first, for the topology of $\widetilde{M}$, one may define $\widetilde{U}\subset\widetilde{M}$ as open $\Leftrightarrow \exists U\subset M$ open with
$$\widetilde{U}=\{(p,o_p)\mid p\in U, o_p\in\{\text{orientations on }T_pM\}$$
Now I'm trying to figure out some chart $(\widetilde{U},\widetilde{\phi})$ at $(p,o_p)$ based on $(\phi, U)$ at $p$. I've tried this:
\begin{align*} \widetilde{\phi}:\widetilde{U}&\to\mathbb{R}^n\\ (p, o_p)&\mapsto \phi(p) \end{align*}
But that obviously doesn't work because it is not even injective. Somehow I have to involve the orientation $o_p$ in the definition, but I really don't know how to do it.
About the orientability, I guess it will have something to do with the orientability of the atlas $\{(\widetilde{U}_{\alpha}, \widetilde{\phi}_{\alpha})\}$, but since I can't figure out the definition of $\widetilde{\phi}$, I'm stuck.
Now for the connectedness of $\widetilde{M}$ and non-orientability of $M$, that I have no idea.
You may want to take a look at John Lee's Introduction to Smooth Manifolds. Chapter 15 contains a fairly comprehensive discussion of orientations. In particular, the section "Orientations and Covering Maps" gives detailed answers to your three questions.