In this paper, we study the local constant and the local linear estimators of the conditional density function with right-censored data which exhibit some type of dependence. It is assumed that the observations form a stationary $α-$mixing sequence. The asymptotic normality of the two estimators is established, which combined with the condition that $\lim\limits_{n\rightarrow\infty}nh_nb_n=\infty$ implies the consistency of the two estimators and can be employed to construct confidence intervals for the conditional density function. The result on the local linear estimator of the conditional density function in Kim et al. (2010) is relaxed from the i.i.d. assumption to the $α-$mixing setting, and the result on the local linear estimator of the conditional density function in Spierdijk (2008) is relaxed from the $ρ$-mixing assumption to the $α-$mixing setting. Finite sample behavior of the estimators is investigated by simulations.