Here's an almost purely set theoretic construction of such a function. I'll describe the graph of a bijection function whose graph is dense in $\mathbb{R}^2$.
Begin by well ordering all the open rectangles of $\mathbb{R^2}$ with order type $\mathfrak{c}$. Also, well order $\mathbb{R}$ with order type $\mathfrak{c}$ as well, so it makes sense to talk of the "least" real number with a given property.
We'll define an injective function inductively. To begin with, choose any point in the first open subset.
Now, assume inductively that for all ordinals $\beta < \alpha$, we have chosen a point $(x_\beta, y_\beta)\in U_\beta$ such that for $\beta\neq\gamma$, we have both $x_\beta\neq x_\gamma$ (so we're still making a function) and $y_\beta \neq y_\gamma$ (so the function is $1-1$).
Consider the open recangle $U_\alpha$. The image of $U_\alpha$ on the $x$-axis is open, so has cardinality $\mathfrak{c}$. Since we have only chosen $<\alpha$ many $x$ values so far and $\alpha < \mathfrak{c}$, the collection of $x$-values we haven't yet picked is nonempty. So, there is a "least" $x$ value we haven't yet chosen. Define $x_\alpha$ to be this $x$ value. Likewise, the image of $U_\alpha$ on the $y$ axis is open, so has cardinality $\mathfrak{c}$, so it contains a $y$ value we haven't picked yet. Define $y_{\alpha}$ to be the "least" such $y$.
Thus, we can continue the induction, so by the principle of transfinite induction, we have now specified a subset $A = \{(x_\alpha, y_\alpha): \alpha < \mathfrak{c}\}\subseteq \mathbb{R}^2$. This $A$ defines a (possibly partial) function by $f(x_\alpha) = y_\alpha$ and by construction this is a function, is 1-1, and has dense graph. What remains is to show that the domain and range are both all real numbers.
To see the domain is all real numbers, pick any real number $x = x_\alpha$. Notice that there are $\mathfrak{c}$ rectangles which, when projected to the $x$ axis, contain $x_\alpha$. But there are only $\alpha$ different $x$s less than $x_{\alpha}$, so for one of these $\mathfrak{c}$ rectangles we picked $x_{\alpha}$. An analogous argument shows the range is all real numbers.