Neuromorphic computing refers to a variety of brain-inspired computers, devices, and models inspired by the interconnectivity, performance, and energy efficiency of the human brain. Unlike the ubiquitous von Neumann computer architectures with complex processor cores and sequential computation, biological neurons and synapses operate by storing and processing information simultaneously with the capacity of flexible adaptation resulting in massive computational capability with much less power consumption. The search for a synaptic material which can closely imitate bio-synapse has led to an alamethicin-doped, synthetic biomembrane which can emulate key synaptic functions due to generic memristive property enabling learning and computation. This two-terminal, biomolecular memristor, in contrast to its solid-state counterparts, features similar structure, switching mechanism, and ionic transport modality as biological synapses while consuming considerably lower power. In this paper, we outline a methodology for using this biomolecular synapse to build neural networks capable of solving real-world problems. The physical mechanism underlying its volatile memristance is explored followed by the development of a model of this device for circuit simulation. We outline a circuit design technique to integrate this synapse with solid-state neuron circuit for hardware implementation. Based on these results, we develop a high level simulation framework and use a training scheme called Evolutionary Optimization for Neuromorphic System (EONS) to generate networks for solving two problems, namely iris dataset classification and EEG classification task. The small network size and comparable to state-of-the-art accuracy of these preliminary networks show its potential to enhance synaptic functionality in next generation neuromorphic hardware.