Bulk and surface state contributions to the electrical resistance of single-crystal samples of the topological Kondo insulator compound SmB6 are investigated as a function of crystal thickness and surface charge density, the latter tuned by ionic liquid gating with electrodes patterned in a Corbino disk geometry on a single surface. By separately tuning bulk and surface conduction channels, we show conclusive evidence for a model with an insulating bulk and metallic surface states, with a crossover temperature that depends solely on the relative contributions of each conduction channel. The surface conductance, on the order of 100 e 2 /h and electron-like, exhibits a field-effect mobility of 133 cm 2 /Vs and a large carrier density of ∼ 2 × 10 14 cm −2 , in good agreement with recent photoemission results. With the ability to gate-modulate surface conduction by more than 25%, this approach provides promise for both fundamental and applied studies of gate-tuned devices structured on bulk crystal samples.