Your question is rather cryptic - at least to me as a non-native English speaker,
I notice that while I charge it, it seems to increase with voltage as well as current
What "increase" do you mean? When a current flows into a capacitor, voltage increases per
V = I*t/C
V = voltage
I = current
t = time
C = capacity
In a typical setup with a voltage source feeding the current to the capacitor the current is limited by the internal resistance (or current limiting) of the voltage source. The limited current will take some time to charge the capacitor. It cannot, however, charge the capacitor to a voltage higher than the voltage of the voltage source.
The current is driven by the difference in voltage from the voltage source and voltage on the capacitor. When you start charging, the voltage on the capacitor is 0 V, therefore the voltage difference is high and current is high, too. As the capacitor gets charged, the voltage on the capacitor increases and the voltage difference diminishes, thus reducing the current,
not increasing it (read e.g. here).
Don't they retain the same amount of voltage and increase the amount of current?
No. As the voltage rises, current is reduced, see above.