We present a comprehensive analysis of interstellar absorption lines seen in moderately-high resolution, high signal-to-noise ratio optical spectra of SN 2014J in M82. Our observations were acquired over the course of six nights, covering the period from ~6 days before to ~30 days after the supernova reached its maximum B-band brightness. We examine complex absorption from Na I, Ca II, K I, Ca I, CH+, CH, and CN, arising primarily from diffuse gas in the interstellar medium (ISM) of M82. We detect Li I absorption over a range in velocity consistent with that exhibited by the strongest Na I and K I components associated with M82; this is the first detection of interstellar Li in a galaxy outside of the Local Group. There are no significant temporal variations in the absorption-line profiles over the 37 days sampled by our observations. The relative abundances of the various interstellar species detected reveal that the ISM of M82 probed by SN 2014J consists of a mixture of diffuse atomic and molecular clouds characterized by a wide range of physical/environmental conditions. Decreasing N(Na I)/N(Ca II) ratios and increasing N(Ca I)/N(K I) ratios with increasing velocity are indicative of reduced depletion in the higher-velocity material. Significant component-to-component scatter in the N(Na I)/N(Ca II) and N(Ca I)/N(Ca II) ratios may be due to variations in the local ionization conditions. An apparent anti-correlation between the N(CH+)/N(CH) and N(Ca I)/N(Ca II) ratios can be understood in terms of an opposite dependence on gas density and radiation field strength, while the overall high CH+ abundance may be indicative of enhanced turbulence in the ISM of M82. The Li abundance also seems to be enhanced in M82, which supports the conclusions of recent gamma-ray emission studies that the cosmic-ray acceleration processes are greatly enhanced in this starburst galaxy.
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