In this work we investigate the behavior of carrier absorption and minority carrier injection in heterojunction solar cells fabricated by spin-coating the organic semiconductor poly(3-hexylthiophene) (P3HT) on n-type crystalline silicon. Using this structure we recently demonstrated a device with open-circuit voltage (V$_{OC}$) of 0.59 V and short-circuit currents (I$_{SC}$) of 22 mA/cm$^2$ at AM 1.5 conditions. In this paper we show, using capacitance-voltage characteristics, that there is a large depletion region in silicon which is responsible for the separation of photogenerated carriers. Furthermore, by measuring minority carrier storage times, we show that the dominant forward-bias dark-current component in these devices is the injection of minority carriers from the anode, through P3HT, in to silicon. This confirms that P3HT functions as a p-type heterojunction contact to silicon that blocks electrons but not holes, explaining the high VOC we observe.