Current flow through capacitor
Current polarity is positive when flowing from V+ to V-.
While -ve electrons flow in the opposite direction of positive current.
  This is the accepted IEE convention.

The peak current is defined by the rate of change of voltage applied limited by some series resistance including that of the capacitor, defined as ESR which is normally neglected except in high pulsed current applications.  

\$I_C = C dV/dt\$

Thus current is maximum with the step voltage applied and declines with dV/dt as the capacitor voltage reaches the applied voltage.

Here shows the capacitor voltage and current.  dV is 10 V but Ic is limited by 10V/1k= 10mA. When the digital sampling time is not small enough, there will be some small error in the peak transient measurement. So it reads only 9.975mA.



Since the voltage source is a 0 Ω "forcing function", the capacitor responds to accept the flow of current through the resistor and charge up towards the applied voltage.