We describe in detail the methods used to obtain the lower bound on the lifetime of neutrinoless double-beta (0νββ) decay in ¹³⁰Te and the associated limit on the effective Majorana mass of the neutrino using the CUORE-0 detector. CUORE-0 is a bolometric detector array located at the Laboratori Nazionali del Gran Sasso that was designed to validate the background reduction techniques developed for CUORE, a next-generation experiment scheduled to come online in 2016. CUORE-0 is also a competitive 0νββ decay search in its own right and functions as a platform to further develop the analysis tools and procedures to be used in CUORE. These include data collection, event selection and processing, as well as an evaluation of signal efficiency. In particular, we describe the amplitude evaluation, thermal gain stabilization, energy calibration methods, and the analysis event selection used to create our final 0νββ decay search spectrum. We define our high level analysis procedures, with emphasis on the new insights gained and challenges encountered. We outline in detail our fitting methods near the hypothesized 0νββ decay peak and catalog the main sources of systematic uncertainty. Finally, we derive the 0νββ decay half-life limits previously reported for CUORE-0, T_1∕2^0ν > 2.7 × 10²⁴ yr, and in combination with the Cuoricino limit, T_1∕2^0ν > 4.0 × 10²⁴ yr.