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Spiking Neural Networks (SNNs) are energy-efficient artificial neural network models that can carry out data-intensive applications. Energy consumption, latency, and memory bottleneck are some of the major issues that arise in machine learning applications due to their data-demanding nature. Memristor-enabled Computing-In-Memory (CIM) architectures have been able to tackle the memory wall issue, eliminating the energy and time-consuming movement of data. In this work we develop a scalable CIM-based SNN architecture with our fabricated two-layer memristor crossbar array. In addition to having an enhanced heat dissipation capability, our memristor exhibits substantial enhancement of 10% to 66% in design area, power and latency compared to state-of-the-art memristors. This design incorporates an inter-spike interval (ISI) encoding scheme due to its high information density to convert the incoming input signals into spikes. Furthermore, we include a time-to-first-spike (TTFS) based output processing stage for its energy-efficiency to carry out the final classification. With the combination of ISI, CIM and TTFS, this network has a competitive inference speed of 2μs/image and can successfully classify handwritten digits with 2.9mW of power and 2.51pJ energy per spike. The proposed architecture with the ISI encoding scheme can achieve ∼10% higher accuracy than those of other encoding schemes in the MNIST dataset.