Recent hardware accelerator designs for machine learning show great promise for allowing complex image recognition tasks to be carried out on resource-constrained platforms. However, the memristor crossbar arrays that many designs rely on are not currently marketed by any commercial manufacturers and must instead be custom-fabricated. The work in this paper was motivated by a desire to build a small, hardware-based neural network that could provide an example for the construction of a physical proof-of-concept device to complement simulated test results for novel neuromorphic circuit designs in cases where custom VLSI design and fabrication are not practicable. Designs based on commercially-available components are proposed for an artificial neuron and synapses, and these designs are employed in the construction of a hardware-based multilayer perceptron that is successfully trained to classify 100-pixel images in handwritten character datasets containing three different image classes.