For 80 d in 2017, the Kepler Space Telescope continuously observed the transitional millisecond pulsar system PSR J1023+0038 in its accreting state. We present analyses of the 59-s cadence data, focusing on investigations of the orbital light curve of the irradiated companion star and of flaring activity in the neutron star's accretion disc. The underlying orbital modulation from the companion star retains a similar amplitude and asymmetric heating profile as seen in previous photometric observations of the system in its radio pulsar state, suggesting that the heating mechanism has not been affected by the state change. We also find tentative evidence that this asymmetry may vary with time. The light curve also exhibits ‘flickering’ activity, evident as short time-scale flux correlations throughout the observations, and periods of rapid mode-switching activity on time-scales shorter than the observation cadence. Finally, the system spent ${\sim } 20\hbox{ per cent}$ of the observations in a flaring state, with the length of these flares varying from <2  min up to several hours. The flaring behaviour is consistent with a self-organized criticality mechanism, most likely related to the build-up and release of mass at the inner edge of the accretion disc.