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 ($Trightarrow P$) 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 $Trightarrow P$ leakage of $<3.8times10^{-4}$ (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 $Trightarrow P$ 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.
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