In this paper, we propose a framework to analyze the performance of buffer-aided (BA) relaying under time-correlated fading channels in terms of the queueing behavior of packets at the relay's buffer. Unlike independent and identically distributed (i.i.d.) fading, correlated fading brings challenges in performance analysis since the state transition probabilities of buffer occupancy become time variant. To overcome this issue, we first establish an aggregate quasi-birth-death (QBD) Markov chain integrating both the buffer occupancy process and the channel fading process, then analyze the stationary distribution of the aggregate chain, and finally extract the stationary distribution of buffer occupancy from it. Using the stationary distribution of buffer occupancy, system throughput, end-to-end delay, and outage probability are derived. Numerical results verify our analyses and show that the throughput of BA relaying under correlated fading channels can approach the one under i.i.d. fading only for loose delay constraints or high fading margins. For stringent delay constraints and low fading margins, correlated fading causes great degradation of throughput. In particular, a throughput loss of about 16% under an infinite buffer (28% under a finite buffer) is observed for the requirement of an average delay of 20 slots and a fading margin of 5 dB. This means that the designs based on i.i.d. fading are not always feasible for correlated fading. According to these observations, some insights on performance degradation and guidelines on redesigning improved policies under correlated fading are provided.