This paper presents a general framework for probabilistic relay selection in asymmetric buffer-aided cooperative relaying systems, which caters to scenarios with both perfect and imperfect channel state information (CSI) during the relay selection (RS) process. The framework extends and generalizes many existing buffer-aided RS schemes. In particular, we introduce an auxiliary stochastic process which assigns varying selection probabilities to different links, considering the dynamic wireless channel and buffer states. Subsequently, we leverage the obtained outage probability and average packet delay (APD) to formulate outage optimization problems while adhering to APD. To address the intricate high-dimensional optimization problems, we employ a deep learning (DL) approach, which involves designing the probability mass function of the auxiliary stochastic process and developing an effective loss function to update the neural network. Simulation results unequivocally demonstrate the superior performance of the proposed DL-based probabilistic relay selection scheme compared to benchmark schemes, particularly in scenarios involving imperfect CSI.