  The Cosmology Large Angular Scale Surveyor (CLASS) is a four-telescope array
observing the largest angular scales ($2 \lesssim \ell \lesssim 200$) of the
cosmic microwave background (CMB) polarization. These scales encode information
about reionization and inflation during the early universe. The instrument
stability necessary to observe these angular scales from the ground is achieved
through the use of a variable-delay polarization modulator (VPM) as the first
optical element in each of the CLASS telescopes. Here we develop a demodulation
scheme used to extract the polarization timestreams from the CLASS data and
apply this method to selected data from the first two years of observations by
the 40 GHz CLASS telescope. These timestreams are used to measure the $1/f$
noise and temperature-to-polarization ($T\rightarrow P$) leakage present in the
CLASS data. We find a median knee frequency for the pair-differenced
demodulated linear polarization of 15.12 mHz and a $T\rightarrow P$ leakage of
$<3.8\times10^{-4}$ (95\% confidence) across the focal plane. We examine the
sources of $1/f$ noise present in the data and find the component of $1/f$ due
to atmospheric precipitable water vapor (PWV) has an amplitude of $203 \pm 12
\mathrm{\mu K_{RJ}\sqrt{s}}$ for 1 mm of PWV when evaluated at 10 mHz;
accounting for $\sim32\%$ of the $1/f$ noise in the central pixels of the focal
plane. The low level of $T\rightarrow P$ leakage and $1/f$ noise achieved
through the use of a front-end polarization modulator enables the observation
of the largest scales of the CMB polarization from the ground by the CLASS
telescopes.
