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This paper investigates the performance of a two-hop decode-and- forward full duplex relaying (FDR) network with a battery-assisted energy harvesting (EH) relay. We consider a practical scenario where the EH relay uses a limited amount of battery energy along with the harvested energy to enhance communication between the source and destination terminals. Utilizing time splitting (TS) protocol for EH, we analyze the performance of a dual-hop FDR system in terms of outage probability and throughput. Further, with fixed battery energy availability, we present closed-form expressions for the TSR parameter that maximizes throughput. For a desired target throughput, we show how the battery energy drawn and the TSR parameter can be chosen so as to extend battery lifetime. Computer simulation results demonstrate accuracy of the derived expressions.