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Heterostructures made from van der Waals materials provide a template to investigate proximity effects at atomically sharp heterointerfaces. In particular, near-field charge and energy transfer in heterostructures made from semiconducting transition metal dichalcogenides (TMD) have attracted interest to design model 2D "donor-acceptor" systems and new optoelectronic components. Here, using of Raman scattering and photoluminescence spectroscopies, we report a comprehensive characterization of a molybedenum diselenide (MoSe$_2$) monolayer deposited onto hexagonal boron nitride (hBN) and capped by mono- and bilayer graphene. Along with the atomically flat hBN susbstrate, a single graphene epilayer is sufficient to passivate the MoSe$_2$ layer and provides a homogenous environment without the need for an extra capping layer. As a result, we do not observe photo-induced doping in our heterostructure and the MoSe$_2$ excitonic linewidth gets as narrow as 1.6~meV, hence approaching the homogeneous limit. The semi-metallic graphene layer neutralizes the 2D semiconductor and enables picosecond non-radiative energy transfer that quenches radiative recombination from long-lived states. Hence, emission from the neutral band edge exciton largely dominates the photoluminescence spectrum of the MoSe$_2$/graphene heterostructure. Since this exciton has a picosecond radiative lifetime at low temperature, comparable with the energy transfer time, its low-temperature photoluminescence is only quenched by a factor of $3.3 \pm 1$ and $4.4 \pm 1$ in the presence of mono- and bilayer graphene, respectively. Finally, while our bare MoSe$_2$ on hBN exhibits negligible valley polarization at low temperature and under near-resonant excitation, we show that interfacing MoSe$_2$ with graphene yields a single-line emitter with degrees of valley polarization and coherence up to $\sim 15\,\%$., version 3, 5 figures