Coded caching is able to exploit accumulated cache size and hence superior to uncoded caching by distributing different fractions of a file in different nodes. This work investigates coded caching in a large-scale small-cell network (SCN) where the locations of small base stations (SBSs) are modeled by stochastic geometry. We first propose a content delivery framework, where multiple SBSs that cache different coded packets of a desired file transmit concurrently upon a user request and the user decodes the signals using successive interference cancellation (SIC). We characterize the performance of coded caching by two performance metrics, average fractional offloaded traffic (AFOT) and average ergodic rate (AER), for which a closed-form expression and a tractable expression are derived, respectively, in the high signal-to-noise ratio region. We then formulate the coded cache placement problem for AFOT maximization as a multiple-choice knapsack problem (MCKP). By utilizing the analytical properties of AFOT, a greedy but optimal algorithm is proposed. We also consider the coded cache placement problem for AER maximization. By converting this problem into a standard MCKP, a heuristic algorithm is proposed. Analytical and numerical results reveal several design and performance insights of coded caching in conjunction with SIC receiver in interference-limited SCNs.