We consider the problem of distributed caching in next generation mobile cellular networks (a.k.a., $5$G) where densely-deployed small base stations (SBSs) are able to store and deliver users' content accordingly. In particular, we formulate the optimal cache allocation policy as a convex optimization problem where a subset of SBSs have their own $i$) local cost function which captures backhaul consumption aspects in terms of bandwidth and $ii$) a set of local network parameters and storage constraints. Given the fact that no coordination involves between SBSs, we then solve this problem distributively using the ADMM approach. The proposed ADMM-based algorithm relies on the application of random Gauss-Seidel iterations on the Douglas-Rachford splitting operator, which results in a low-complexity and easy-to-implement solution for SBSs. We examine the convergence of our proposed algorithm via numerical simulations with different parameters of interest such as storage capacity distribution of SBSs, content catalogue size, demand intensity and demand shape. Our numerical results show that the proposed algorithm performs well in terms of convergence and requires less iterations as the number of contents in the catalogue increases.