This study presents a framework to analyze the performance of uplink localization with reconfigurable intelligent surfaces (RISs) in large-scale cellular networks. First, we propose a novel RIS-aided uplink localization algorithm, where the received signal strength (RSS) is observed at the base station (BS) for various pre-defined phase shift patterns of the RIS, i.e., a codebook of beams. We present a maximum likelihood estimator (MLE) and evaluate its performance by comparing it to the position error bound (PEB), defined as the square root of the Cramér-Rae lower bound (CRLB). Then, to analyze the localization performance on a large scale, we employ stochastic geometry tools, allowing the derivation of a tractable expression for the marginal PEB distribution. The obtained results demon-strate that the proposed algorithm converges to the CRLB for a narrow search grid, in a high SNR regime. Furthermore, higher BS density, number of RIS elements, and RIS element size are shown to enhance localization precision.