With most off the shelf meters, the meter won't detect much current because the capacitor essentially gets shorted when the meter is placed across it.
With a capacitor the exponential timeconstant (~60% of the initial voltage) would be
$$ \tau = RC$$
With a meter in current mode most likely having a resistance of lower than 0.1Ω
$$ \tau = (0.1Ω)(1000uF)= 100us$$
So this means in about 100us most of the voltage fades away
Source: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imgele/capdis.gif
The current could also be calculated with the equation above, but you must know how much voltage \$V_0\$ you charged the capacitor to.
The current at 100us and 5V would be less than 19mA, in another few hundred us (like 1000us) the current from a 1000uF cap would be less than 1mA which would be hard to see for many meters because of the short duration of the current.
The capacitor you constructed probably has a smaller value of capacitance (maybe lower than 1uf) so the time would be even shorter than 100us.
So, one way to overcome this would be to put a resistor in series with the capacitor to make the time constant longer and/or use a really good current meter that can measure. Another good capability to have would be graphing. Don't get me started on fitting exponential curves (big hint).