The j-invariant as a modular function is typically defined $$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$ since $E_4$ is a modular form of weight 4 and $\Delta$ has weight 12, it follows that $j$ is a modular function.

My first question: Is there a different derivation of the $j$-invariant? Perhaps one using the geometry of the modular curve $X(1) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}^*$ or one more algebraic in nature? I.e., if I wanted to construct a function that is $\mathrm{SL}_2(\mathbb{Z})$-invariant, how should I start?

For my second question, There is the well known result that the field of all modular functions is equal to $\mathbb{C}(j)$. This is typically using the q-expansions of $j$ and a residue like theorem for modular forms.

However, $X(1) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}^*$ has a model so that $X(1)(F)$ parametrizes $\bar{F}$-isomorphism classes of elliptic curves defined over $F$. So, one might expect that there is an algebraic proof that the $j$-invariant generates all modular functions that works over any field. Does such a proof exist?