Since that the answer to this question will depend on the particular semantics that I'm using, allow me to first define truth of $\Box\varphi$ and $\Diamond\varphi$ at a world $\alpha$ in a model $\mathfrak{M}=(\mathit{W},\mathit{R},\mathit{P})$, where $\mathit{R}$ is a preorder and $\mathit{P}$ is a map from the set of atomic sentences of the language to $\mathcal{P}(\mathit{W})$ (the power set of $\mathit{W}$).
Note: I'm pretty sure this it what most people would simply call a Kripke model, but I thought that to avoid misunderstandings it would be better to be completely explicit.
$\bullet$ $\vDash_{\alpha}^{\mathfrak{M}}\Box\varphi$ if and only if for all $\beta$, $\alpha\mathit{R}\beta$ implies $\vDash_{\beta}^{\mathfrak{M}}\varphi$.
$\bullet$ $\vDash_{\alpha}^{\mathfrak{M}}\Diamond\varphi$ if and only if there exists $\beta$ such that $\alpha\mathit{R}\beta$ and $\vDash_{\beta}^{\mathfrak{M}}\varphi$.
Note: Again, I think this is just Kripke semantics, but I wanted to make it explicit for clarity.
Now, if I wanted to write the truth sets corresponding to these formulae in the model $\mathfrak{M}$, I could just define
$\bullet$ $\Vert\Box\varphi\Vert^{\mathfrak{M}}=\lbrace \alpha\in\mathit{W}\mid(\forall \alpha\mathit{R}\beta)(\beta\in\Vert\varphi\Vert^{\mathfrak{M}})\rbrace$
$\bullet$ $\Vert\Diamond\varphi\Vert^{\mathfrak{M}}=\lbrace \alpha\in\mathit{W}\mid(\exists \alpha\mathit{R}\beta)(\beta\in\Vert\varphi\Vert^{\mathfrak{M}})\rbrace$
The thing is I don't find this completely satisfying, since the truth sets of the formulae $\neg\varphi$, $\varphi\wedge\psi$, $\varphi\Rightarrow\psi$, etc, can be written in terms of elementary operations involving the truth sets of $\varphi$ and $\psi$. I've been trying for a while to find a more compact way to treat the case of the modal operators, but there doesn't seem to be any way other than what I just did. I thought I could pull it off using upsets, but I'm not entirely sure about that approach.
Anyway, please let me know if there's a nicer way to describe $\Vert\Box\varphi\Vert^{\mathfrak{M}}$ and $\Vert\Diamond\varphi\Vert^{\mathfrak{M}}$ in terms of $\Vert\varphi\Vert^{\mathfrak{M}}$. Maybe I'm just wasting my time, but my gut tells me there should be a nice compact way to do it.
Thank you all in advance.
This can't be done in any good sense I can think of.
The issue is that modalities take into account the relational aspect of a Kripke model in a way that connectives don't. For example, if I know the set of worlds $W$ and the truth set $\Vert\varphi\Vert$, I can figure out the truth set $\Vert\neg\varphi\Vert$: it's just $W\setminus\Vert\varphi\vert$. However, $W$ and $\Vert\varphi\Vert$ alone are not enough to determine either $\Vert\Diamond\varphi\Vert$ or $\Vert\Box\varphi\Vert$: contrast e.g. $R=W^2$ and $R=\emptyset$.
So unlike the connectives, we can't describe how modalities affect truth-sets without invoking $R$. And once we bring $R$ into the picture, the standard descriptions are really as simple as we could hope for.