A controlled ambipolar-to-unipolar (n-type) conversion, along with a maximum fourfold increase in the electron mobility, in graphene field-effect transistors (FETs) is achieved by coating the surface of graphene with a layer of a mixed polymer system, poly(ethylene imine) (PEI) in poly(ethylene glycol) (PEG). The PEG serves as a physisorption adhesion agent for the PEI. Both unipolar and ambipolar n-type doping can be realized by adjusting the thickness of PEI films atop the graphene channel. The observed phenomena are attributed to the doping/dedoping effects of the external PEI film. The study provides a guide to engineering graphene transport properties through chemical modifications.