Perhaps my post led to a misunderstanding. I did not specify the dimension of the projective space; for a surface of high genus the embedding is initially into a projective space of high dimension. For a surface of high genus you You will need many independent enough meromorphic functions to realize it as an embedded nonsingular subvariety. This you can get by taking various rational functions of two such meromorphic functions analogous to pe and pe'. Afterwards getting it into a projective space of low dimension is, as others have pointed out, an issue of algebraic geometry. Also, the meromorphic functions I talk about are meromorphic functions on the Riemann surface; i. e. holomorphic mappings from the surface into the Riemann sphere a.k.a. the extended complex plane a.k.a. one-dimensional complex projective space.
But Shafarevich explains this much better than I.
The question of explicit embedding obviously depends on how the Riemann surface is actually given to you concretely. If it is given by a Fuchsian group acting on the unit disk, you can embed it using Poincare series. These are analogues of the series used to define the Weierstrass pe-function.