Hom is right adjoint to tensor product. Have you tried to write the forgetfull functor as $F(M)=R\otimes_R M$?
I didn't check the details, but I think you can do something like
$$Hom_k(Forget(M),W)=Hom_k(M,W)\cong Hom_k(R\otimes_RM,W)\cong Hom_R(M,Hom_k(R,W))$$ where $M$ is an $R$-module, $W$ is a vector space, and the $R$-structure on $Hom(R,W)$ is from the right structure of $R$, that is $(r\cdot f)(x)=f(xr)$.