In his paper cited above on Abelian functions, and appealing to his earlier thesis results, Riemann sketches a functor from the category of irreducible plane algebraic curves with rational maps and rational functions, to compact connected complex one manifolds with holomorphic maps and meromorphic functions. [One compactifies the curve as a projective curve, and then desingularizes it as a manifold. Rational maps become holomorphic on the desingularization because of the Riemann extension theorem. Much of this occurs in the book of Miranda.] Riemann then proves that the field of rational functions on the plane curve equals the field of meromorphic functions on the manifold.
Indeed, he shows that if there is a single non constant meromorphic function on the manifold say of degree n, such as one of the plane variables gives, then the entire field of meromorphic functions on the manifold is algebraic of degree at most n over the field obtained by adjoining this one function to the constants. It follows that all holomorphic maps of the manifolds arise from rational maps of the curves and that in particular holomorphic equivalence of the manifolds is the same as birational equivalence of the curves.
This implies that your construction of a (non unique) plane curve from a function field, always yields birationally equivalent curves, hence isomorphic Riemann surfaces. Riemann himself considers the problem of birational equivalence in his paper and determines the lowest degree of a plane polynomial representation for a given Riemann surface in terms of the lowest degree of a map from that surface to the Riemann sphere.
All this is actually proved essentially rigorously in his paper, appealing only to his extension theorem for holomorphic functions. What has been criticized as to rigor is the inverse correspondence that all compact connected complex one manifolds arise from plane curves. The method was to produce harmonic functions in plane regions by the Dirichlet principle, which method was justified by Hilbert and others later, as recorded in the books of Weyl and Siegel and Springer. More modern approaches occur in Gunning, and the article by Cornalba in his Trieste lectures.
As noted above, for higher dimension there exist compact complex manifolds not arising from algebraic varieties, and there exist such examples in Shafarevich, e.g. of compact complex tori which do not have meromorphic function field of the correct transcendence degree. Manifolds which do have such meromorphic function fields, and hence could be algebraic, are called Moishezon manifolds, and he showed they can always be blown up to become algebraic, if I recall correctly.
In that famous Abelsche Functionen paper, Riemann goes on to deduce rigorously his famous inequality, by estimating the rank of a period matrix, assuming only the existence of sufficient meromorphic one forms of 1st and 2nd kinds, i.e. either holomorphic, or having zero residues at every pole. Although his proof of the existence of these forms in the manifold setting relies on his disputed use of the Dirichlet principle, he remarks in section 9 of the paper that one can simply write them down in the case of plane curves, and he actually does so for the holomorphic ones, using the "Poincare" residue principle. He says he could write down the others as well, but will not stop to do so. Such explicit expressions are given in the book on Plane Algebraic Curves of Brieskorn, by way of showing how to represent all cohomology classes on a curve by meromorphic forms of 1st and 2nd kinds. E.g. on the cubic curve y^2 = x(x-1)(x-t), the form x(x-1)dx/y^3 is an elementary form of 2nd kind with one double pole (at (t,0)) but zero residue.
If one grants that Riemann knew how to do this, as he said, then the foundation for his proof of the Riemann inequality is completely provided, and at least for plane curves, there is no need for Hilbert's analytic foundations to bolster Riemann's argument in the complex algebraic case. The 1865 paper of Roch, in which he completes Riemann's argument, rests solely on Green's theorem to compute Riemann's period matrix as a residue integral, hence is completely solid. 17 years later, Brill and Noether, using the same matrix computed by Roch, apparently showed that one can exploit the duality between divisors of form D and K-D to actually give the full proof using only the existence of the integrals of 1st kind. Since that paper was so influential, Roch's residue matrix (occurring in the middle of the second page of his paper) is now usually known as the Brill Noether matrix.
In addition to the functor from curves C to one manifolds X, Riemann also considered two more functors, the symmetric products X^(d) and the Jacobian variety J(X), as well as a natural transformation between them X^(d)--->J(X), called the Abel map. The fibers of this map are the linear series |D|, ("Abel's theorem"), and the derivative of this map is the Roch ("B-N") matrix, (by the fundamental theorem of calculus). Hence the Riemann Roch theorem becomes the assertion that the fibers of the Abel map are non singular as schemes. I.e. the fiber dimension dim |D|, equals the dimension of the kernel of the derivative, d-g+h^0(K-D). This is the formulation of Mattuck and Mayer.
Best Answer
This deep fact is essentially the same as the uniformization theorem. The problem is how to construct at least one holomorphic or meromorphic form with prescribed singularity. All known proofs use some Analysis, and none of them is simple. Once you have Uniformization, it is easy to construct holomorphic forms.
A good modern proof (in full generality) is contained in the book of J. Hubbard, Teichmuller Theory. The complexity of the proof depends on how exactly you define a Riemann surface. A Riemann surface is a 1-dimensional complex manifold. A 1-dimensional complex manifold is separable (has a countable base). This fact is a part of the modern statement of the Uniformization theorem. However, if you include this separability to the DEFINITION of a Riemann surface, the proof substantially simplifies. Especially simple proof (assuming separabiity) can be found in Goluzin's book Geometric theory of functions of a complex variable.
All Riemann surfaces arising in real life are easily seen to be separable (give me an example if I am wrong), so there is no real harm if we include this in the definition:-)
But with or without this separability condition, all proofs of existence of holomorphic forms or of the uniformization use Analysis. Complex or real, I don't see a sharp distinction between these two.