Frequency vs. Amplitude modulation in a circuit?
Is amplitude of the waves modulated by the amount of current?


The amplitude of the current signal is half of the difference between the maximum and minimum of the current in each cycle of the oscillation.

For example, if the current over time is described by the equation

$$i(t) = A \sin(\omega{}t+\phi)$$

then A is the amplitude of signal because the current oscillates between +A and -A (and A has units of amps).

In the circuit shown, there is nothing modulating the current. Modulation happens when some other signal (like an audio waveform) changes the amplitude (for example) of the current waveform. This would require a more complex circuit than what is shown in your example.


  Frequency is modulated by the frequency of the capacitor release of energy, correct? 


Again, your circuit shows no mechanism for modulating the frequency. If you used a variable capacitor and the capacitance was controlled by another signal, that would cause frequency modulation. But it would probably also cause some undesirable parasitic amplitude modulation. To see some typical frequency modulator circuits, you can do a google image search for "frequency modulator" and look at the schematics that are found.


  That would mean that using a capacitor with a lower farad rating should produce a higher frequency?


Generally the resonance of an LC tank circuit like in your example is given by 

$$\omega_0=\frac{1}{\sqrt{LC}}$$

So the oscillation frequency can be increased by reducing either the capacitance or inductance value.


  How would one go about keeping a sustained current to the inductor since it's constantly losing current without messing up the cycle?


You can use an oscillator circuit. Generally this means adding some kind of amplifier to the circuit to add energy to the waveform as quickly as it is lost to parasitic resistance and to feeding the load.