How to detect a simple closed curve from the element in the fundamental group?

There is no simple necessary and sufficient condition for whether an element of the fundamental group can be realized by a simple closed curve. However, there are a variety of algorithms known. I believe that the first such algorithm is given in

Reinhart, Bruce L. Algorithms for Jordan curves on compact surfaces. Ann. of Math. (2) 75 1962 209–222.

But probably the most used one is the one in

Birman, Joan S.; Series, Caroline, An algorithm for simple curves on surfaces. J. London Math. Soc. (2) 29 (1984), no. 2, 331–342.

It uses hyperbolic geometry.

I should remark that both of these papers concern unbased curves; however, if $\Sigma_g$ is a closed genus $g$ surface, then an element $\gamma \in \pi_1(\Sigma_g,\ast)$ can be realized by a based simple closed curve if and only if it is freely homotopic to a simple closed curve. Indeed, if $\gamma$ is freely homotopic to a simple closed curve, then there is some $x \in \pi_1(\Sigma_g,\ast)$ such that $x \gamma x^{-1}$ can be realized by a based simple closed curve. But it follows from the Dehn-Nielsen-Baer theorem that inner automorphisms of $\pi_1(\Sigma_g,\ast)$ can be realized by based homeomorphisms of the surface, so thus $\gamma$ can also be realized by a based simple closed curve.

Various algorithms for determining whether a given conjugacy class contains a simple representative are given in the following papers.

Reinhart, Bruce L., 'Algorithms for Jordan curves on compact surfaces', Ann. of Math. (2) 75 1962 209–222.

Zieschang, Heiner, 'Algorithmen für einfache Kurven auf Flächen', (German) Math. Scand. 17 1965 17–40.

Birman, Joan S; Series, Caroline, 'An algorithm for simple curves on surfaces', J. London Math. Soc. (2) 29 (1984), no. 2, 331–342.

A recent paper by Patricia Cahn, A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface gives a strengthening of Turaev's bracket. She shows that a related invariant of a free homotopy class is 0 if and only if the class is a power of a simple loop. This answers your question, modulo determining if your class is a proper power of some other element; I have no idea how difficult this latter is, or indeed how calculable her invariant is. There are some worked examples in the paper.