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We present the most intensive, high-quality spectroscopic monitoring of optical Wolf-Rayet emission lines ever obtained. The Wolf-Rayet star HD 192103 (=WR 135; subtype WC8) was observed in the 5650-5840 Å regime alternately from both the William Herschel Telescope and the Canada-France-Hawaii Telescope. The final data consist of a series of 197 spectra spread over 64 hr, each with a resolving power λ/Δλ≃20,000 and a signal-to-noise ratio in the continuum ≃450 per 3 pixel resolution element. We clearly and unambiguously identify stochastic, structured patterns of intrinsic variability at the 1%-2% level of the line flux in the broad C III λ5696 emission line. The λ5801/12 doublet emission is also found to be variable at the 0.2%-0.5% level of the line flux. We find a correlation between the variability patterns observed in C III and C IV, which suggests a significant overlap in the emission volumes of these transitions, although C IV is known to arise somewhat closer to the star. We attempt to reproduce the observed line profile variation patterns using a simple phenomenological model, which assumes the wind to be fully clumped. With a minimal set of assumptions, we are able to reproduce both the shape and the variability in the C III λ5696 emission profile. We show that the variability pattern provides constraints on the radial extent of WR 135's wind where C III is produced, as well as on the local wind acceleration rate. However, our simple clump model does not reproduce the lower variability in the C IV doublet unless we assume the C IV emission to occur in a much larger volume than C III, implying that significant C IV emission occurs farther out in the wind than C III. We suggest that while some C IV emission might occur farther out, possibly because of reionization from shocks, a more likely explanation is that wind clumping significantly increases with distance from the star, leading to larger variability levels in C III, formed farther out than most of C IV. Alternatively, optical depth effects and/or local ionization gradients within clumps could conspire to attenuate clumping effects in the C IV emission line while enhancing them in the C III line.