How would I go about converting a change in capacitance into a voltage?
The easiest way that I can think of is using that capacitor in an oscillator circuit. Unless your capacitor is very small in value (less than 100 pF) you usually do not need to use an inductor to use that capacitor in an oscillator. An oscillator can also be made with an RC circuit, this is used in the 555 timer chip.

The frequency of oscillation would change as the capacitor changes value. Then you only need to measure the frequency of oscillation which is is often as easy as counting pulses for a certain time (like 1 second) and then displaying how many cycles were counted.

Please state between what values your capacitor will vary so we can give you more practical advise in the direction of a usable solution.