The Kepler Mission revolutionized exoplanet science and stellar astrophysics by obtaining highly precise photometry of over 200000 stars over 4yr. A critical piece of information to exploit Kepler data is its selection function, since all targets had to be selected from a sample of half a million stars on the Kepler CCDs using limited information. Here we use Gaia DR2 to reconstruct the Kepler selection function and explore possible biases with respect to evolutionary state, stellar multiplicity, and kinematics. We find that the Kepler target selection is nearly complete for stars brighter than Kp<14mag and was effective at selecting main-sequence stars, with the fraction of observed stars decreasing from 95% to 60% between 14<Kp<16mag. We find that the observed fraction for subgiant stars is only 10% lower, confirming that a significant number of subgiants selected for observation were believed to be main-sequence stars. Conversely we find a strong selection bias against low-luminosity red giant stars (R~3-5R_{sun}_, Teff~5500K), dropping from 90% at Kp=14mag to below 30% at Kp=16mag, confirming that the target selection was efficient at distinguishing dwarfs from giants. We compare the Gaia Re-normalized Unit Weight Error (RUWE) values of the observed and nonobserved main-sequence stars and find a difference in elevated (>1.2) RUWE values at ~{sigma} significance, suggesting that the Kepler target selection shows some bias against either close or wide binaries. We furthermore use the Gaia proper motions to show that the Kepler selection function was unbiased with respect to kinematics.