In the affine case, there is a sweet way to define the first Chern class as follows:
Let $R$ by the coordinate ring and $M$ the $R$-module correspond to our sheaf. As $M$ is torsion-free, one can embed $M$ in to a free module:
$ 0\to M \to F \to N \to 0$
( you need $M$ to be of constant rank, and that rank would be the rank of $F$). In this $N$ would be torsion, so the support is finite. Take the cycle $c(N) = \sum length(N_p)[p]$ where $p$ runs over the support of $N$. Then define $c(M)= -c(N)$.
In general, one could get codimension 1 cycles by picking them from any prime filtration of $M$(you needs to show that what you get from 2 different filtrations are rationally equivalent). This paper contains a treatment of that result.
More general setting
The question actually fits into the more general setting that is described in David Ben-Zvi's answer to this MO question (which is about categorification of the Chern character).
HRR for dg algebras
Shklyarov is the one who developped RR theorem for noncommutatve derived schemes (by this one should understand smooth proper DG algebras): http://arxiv.org/abs/0710.1937.
Shklyarov's result has been improved recently by Petit, Lunts, and also Polishchuk-Vaintrob in the context of matrix factorizations.
Let me explain what the statement is. Let $A$ be a proper and homologically smooth dg algebra $A$ (proper means that $\sum_n dim(H^n(A))<+\infty$, and homologicaly smooth means that $A$ is perfect in $D(A\otimes A^{op})$). Let $M$ be a perfect $A$-module. There is a trace map $ch:Hom_{D^{perf}(A)}(M,M)\to HH_0(A)$ (see e.g. this paper of Caldararu-Willerton for a very nice description of the kind of traces I am speaking about), which you can consider as being the Chern character.
Now for an $A$-module $M$ and an $A^{op}$-module $N$ we can consider the $k$-module $N\otimes_AM$ (all my tensor products are derived). Then for $f:M\to M$ and $g:N\to N$ we can consider $ch(g\otimes f)=str(g\otimes f)\in HH_0(k)=k$. Finally the formula is
$$
ch(g)\cup ch(f)=ch(g\otimes f)
$$
where $\cup:HH_*(A^{op})\otimes HH_*(A)\to HH_*(k)=k$ is the so-called (categorical) Mukai pairing. This is actually more a Lefshetz type formula.
The Todd class is actually hidden in the Mukai pairing (the point is that for associative algebras there is no analogon neither for the Todd class, nor for the usual pairing given by integration).
RR for D-modules
To my knowledge the first one who proved a RR Theorem for D-modules is Laumon (Sur la categorie derivee des D-modules filtres, Algebraic Geometry, M. Raynaud and T. Shioda eds, Lecture Notes in Math. Springer-Verlag 1016 pp. 151–237, 1983). Then Schapira and Scheinders also considered it (Index theorem for elliptic pairs II. Euler class and relative index theorem, Asterisque 224 Soc. Math. France, 1994) and made a very important conjecture which has been proved by Bressler-Nest-Tsygan using methods of deformation quantization ( http://arxiv.org/abs/math/9904121 and http://arxiv.org/abs/math/0002115) developped by Fedosov.
There is also a paper of Engeli and Felder that gives a Lefschetz type formula. Their approach has been later clarified by Ramadoss (he has many paper on this subject that you can find on arXiv).
The subject really moved to deformation quantization. You can learn a lot about all this (with also more details on who one should credit for what) in Section 4, 5 and 6 of this book by Kashiwara and Schapira.
If you have a formal noncommutative deformation $A_\hbar$ of the structure sheaf $\mathcal O_X$ (maybe as a twisted resheaf, or algebroid stack - this is what Kashiwara and Schapira call a DQ algebroid), then you can play the same game as with a smooth proper dg algebras : define a trace with values in Hochschild homology, and state a HRR type Theorem (better, Lefshetz formula) about the compatibility of the cup product on Hochschild homology with the composition of kernels. The reason for that relies on some finiteness and duality properties for cohomologicall complete $A_\hbar$-modules. The main difficulty is then to prove such a result.
To conclude this paragraph, let me observe that (the Rees algebra of) $\mathcal D_X$ can be viewed as a deformation of $\mathcal O_{T^*X}$.
Relation between the question for nc schemes and D-modules
Last but not least, the relation between the derived non-commutative geometry and deformation quantization stuff is adressed in a very recent preprint of Petit: his strategy to prove HRR for DQ algebroids is to use the result for smooth proper DG algebras. Namely, he proves that some derived category of cohomologically complete modules over a DQ-algebroid on a projective variety has a compact generator.
This subject is currently very active.
I apologize for that this answer reduces to a (non-exhaustive) list of references. If you have a more specific question I can tell you where to go in these references in order to (hopefully) find an answer.
Best Answer
Yes. There is a Riemann-Roch for smooth projective curves over arbitrary fields. It was proved by the German school of function fields in the 30's. From (2) I deduce that you've been reading Weil's "Basic Number Theory". Anyway, the proof that Weil gives there is a shortened version of a proof he gave of the full theorem. It's in his collected works ("Algebraische Beweis der Riemann-Roch Satz", or some such title). The proof is reproduced in many books, particularly those with "function field" in the title (e.g. Stichtenoth) or Lang's "Introduction to algebraic and abelian functions" (watch for misprints, as usual). You can also prove it with the modern geometric machinery. A book that bridges the two approaches is Serre's "Groupes algebriques et corps de classes". There is absolutely no restriction on the ground field, except the proof is slightly more tortuous when the field is not perfect.