How could a hobbyist astronomer determine apparent magnitude of a star?
The easiest way to determine the magnitude of a given star is probably to use the Pogson relation. The idea is to determine the magnitude of a star knowing the magnitude of a reference star; it is thus quite easy, using a well-known reference as Vega or Sirius.

The Pogson relation is given by:

$$m_1-m_2=-2.5\ log\ \left({\frac{E_1}{E_2}}\right)$$

where $m_1$ and $m_2$ are the magnitudes of star 1 and star 2 (your reference star), and $E_1$ and $E_2$ brightness (that could be in arbitrary units, which is a good news for you).

With a CCD detector the brightness is easy to determine since it's just the flux of a pixel. You don't have to convert it in physical units (as Watt per square meter) since you have a ratio $\left({\frac{E_1}{E_2}}\right)$, so the raw intensity is enough to determine the magnitude.

As for any other measurement with a CCD camera, you have of course to determine the bias, dark and flat-field and to reduce your data before estimating any magnitude. The precision will depend on the number of observations performed and, as always, it is better to perform several short exposure images than a single long exposure one.