Recently, cache-enabled heterogeneous cellular networks (C-HetNets) have been emerged as a promising solution for next-generation wireless networks to cope with exponentially increasing demands for high data rate and low latency multimedia services. In this paper, we design both the caching and the delivery policies in the downlink of an orthogonal frequency division multiple access (OFDMA)-based C-HetNet which operates in two phases: caching phase and delivery phase. In the caching phase, we propose novel transmission-aware caching policies by coupling the cache placement strategy and physical-layer transmission to exploit the benefits of flexible physical transmission opportunities for access and backhaul links, simultaneously. We specifically apply a joint optimization of cache placement and ergodic radio resource allocation for both the access and backhaul links to extend the flexibility of the cache placement strategies and cover each time period of the delivery phase. Since most of the current research efforts in cache-assisted wireless networks neglected the fair treatment of mobile users (MUs), in this work, we propose two schemes called proportional fairness and min-max fairness schemes. In the proportional fairness scheme, we minimize the total weighted latency of MUs while in the min-max fairness scheme, we minimize the maximal latency of MUs in each cell. In numerical assessments, we investigate the effect of the proposed fairness and transmission-aware caching policies compared to the conventional proactive caching strategies which are only based on the contents popularity and storage capacities.