This paper introduces Data-enabled Predictive Control Hyperparameter Tuning via Differentiable Optimization (DeePC-Hunt), a backpropagation-based method for automatic hyperparameter tuning of the Data-enabled Predictive Control (DeePC) algorithm. The necessity for such a method arises from the critical importance of hyperparameter selection to achieve satisfactory closed-loop DeePC performance. The standard methods for hyperparameter selection are to either optimize the open-loop performance, or use manual guess-and-check. Optimizing the open-loop performance can result in unacceptable closed-loop behavior, while manual guess-and-check can endanger the plant. DeePC-Hunt provides an alternative method for hyperparameter tuning which uses an approximate model of system dynamics and backpropagation to directly optimize hyperparameters for the closed-loop performance of DeePC. Numerical simulations demonstrate the effectiveness of DeePC in combination with DeePC-Hunt in a complex stabilization task for a nonlinear system and its superiority over model-based control strategies in terms of robustness to model misspecifications.