This paper characterizes an achievable channel coding rate for a noiseless binary communication channel with an energy harvesting (EH) transmitter at a given blocklength n and error probability ε. As energy arrives randomly at the transmitter, codewords must obey the cumulative stochastic energy constraints. The coupling of the energy constraints on the symbols in a codeword makes the analysis fundamentally different from that of discrete memoryless channels. We first adopt a random coding scheme to construct the codebook with statistical information of the EH process. We then analyze the statistics of the corresponding output sequence. Specifically, we prove that the average number of mismatches between the input codeword and the output sequence scales as O(√n). Based on such characterization, we then propose a decoding scheme, and analyze the corresponding probability of decoding error. Finally, we explicitly characterize the maximum size of the length-n codebook generated by the random coding scheme in order to achieve the average probability of error ε. This leads to a lower bound on the maximum achievable channel coding rate for the EH communication channel. We show that the gap between the lower bound and the corresponding channel capacity under an equivalent average power constraint scales in O(llog n/√n), where l is a constant depending on the error probability ?, and the statistics of the energy harvesting process.