It really depends on what counts as a description. e.g. for some purposes, "the algebraic closure of $\mathbb{C}(x)$" is already a rather good and easy description!
For explicit computation, it's often good enough to just construct extension fields of $\mathbb{C}(x)$ as you need them -- e.g. to start, you might decree that $\alpha$ satisfies $f(x, \alpha) = 0$, and then continue by working in the field $\mathbb{C}(x, \alpha)$ (which is, of course, isomorphic to $\mathbb{C}(x)[y] / f(x, y)$.
As a twist on this, there's probably some way to use Riemann surfaces (together with a local choice of branch) embedded in $\mathbb{C}^2$ to name elements of the algebraic closure, but it's not immediately obvious to me if it would work.
The field of complex Puiseaux series is not an algebraic closure -- it is the algebraic closure of $\mathbb{C}((x))$ and thus "too big" -- but it does contain an algebraic closure of $\mathbb{C}(x)$, so you can name things this way.
As an aside, writing things like $\sqrt[n]{x}$ is tricky, since the algebraic closure actually contains $n$ elements that can rightfully claim that name, and without choosing a specific representation of the set, it is impossible to specify which of those $n$ elements you mean. (Of course, you can get around this by simply decreeing that we choose one before-hand, and use $\sqrt[n]{x}$ to refer to it)