A 256-pixel camera for a 4-meter class telescope based on Imaging Atmospheric Cherenkov Technique (IACT) is under development. It would be mounted at the focal plane of TeV Atmospheric Cherenkov Telescope with Imaging Camera (TACTIC) array element at Mount Abu (Rajasthan, India). The telescope would be used for detection of very high energy gamma rays of celestial origin. Instead of the traditional choice of photo-multiplier tube, the camera employs silicon photo-multiplier (SiPM) as pixel sensor. The front-end electronics of the camera processes signals from 256 pixel sensors. The back-end electronics samples the processed pixel signals, generates event triggers and records the sampled signals. The critical design objectives to be met by the back-end electronics include large dynamic range of photon detection, capability of in-situ gain calibration for each pixel chain, low chance trigger rate, recording the pixel signal profile with a sampling rate of 1 Giga Samples Per Second (GSPS), low dead time (<1 millisecond) and run-time control of the camera operation. Also, the camera cost, power, size and weight should be as low as possible. The important back-end tasks like pixel signal processing, generating pre-triggers, digitization of the pixel pulse profiles, packing and sending the digitized data are performed by the custom designed Cluster Digitizer Module (CDM). CDM is a 16-channel readout system, and its design revolves around an ultra-fast analog sampler chip and a set of FPGAs. The CDM is thoroughly tested in the lab using test signals from SiPM as well as high performance function generator. The paper describes the design, development and performance of the prototype CDM.