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The COS Legacy Archive Spectroscopic SurveY (CLASSY) is designed to provide the community with a spectral atlas of 45 nearby star-forming galaxies that were chosen to cover similar properties to those seen at high z (z > 6). The prime high-level science product of CLASSY is accurately coadded UV spectra, ranging from ∼1000 to 2000 Å, derived from a combination of archival and new data obtained with HST's Cosmic Origins Spectrograph (COS). This paper details the multistage technical processes of creating this prime data product and the methodologies involved in extracting, reducing, aligning, and coadding far-ultraviolet and near-ultraviolet (NUV) spectra. We provide guidelines on how to successfully utilize COS observations of extended sources, despite COS being optimized for point sources, and best-practice recommendations for the coaddition of UV spectra in general. Moreover, we discuss the effects of our reduction and coaddition techniques in the scientific application of the CLASSY data. In particular, we find that accurately accounting for flux calibration offsets can affect the derived properties of the stellar populations, while customized extractions of NUV spectra for extended sources are essential for correctly diagnosing the metallicity of galaxies via C iii] nebular emission. Despite changes in spectral resolution of up to ∼25% between individual data sets (due to changes in the COS line-spread function), no adverse affects were observed on the difference in velocity width and outflow velocities of isolated absorption lines when measured in the final combined data products, owing in part to our signal-to-noise regime of S/N < 20.