For the Maths of calculating uncertainty the standard document is the GUM. Which describes all the maths but can be somewhat unclear if you don't already have some idea how it is supposed to work.
Depending on your current level of expertise there are several good introductions to uncertainty calculations. I would recommend A Beginner's Guide to Uncertainty of Measurement, which is freely downloadable.
For any uncertainty calculation you need a model of your output given the inputs. For your case a very simple model would be purely linear, with distance proportional to voltage measured.
$$L=\alpha V$$
In this simple case, for fixed $\alpha$, the uncertainty in your distance is $\alpha u_v$ but more generally for a function $f$ with various inputs $x_i$.
$$u^2=\sum_i (\frac{df}{dx_i})^2 u_{x_i}^2 $$
Your uncertainties in V (or your general $x_i$) can be determined either using prior knowledge or by repeated measurement (at a given distance or similar).
Calibration of your system is necessary to determine any coefficients in your model, such as $\alpha$, and to help inform the choice of model itself. The necessary calibration will vary in a case by case basis, but for your setup I guess you will want to measure a series of known distances and plot a calibration curve of distance against voltage.
It is important to note that as you have uncertainty in your voltage and in the distance of your standard lengths used for calibration, as well as other non-linearities no there will inevitably be some uncertainty in your calculation of $\alpha$, unlike in my simple example. This will add some additional uncertainty proportional to the voltage to your measurement.
