The extremal dependence of defaults, and negative correlation between defaults and their recovery rates, are of major interest in modeling portfolio credit risk. In order to incorporate these two features, we propose a portfolio credit risk model with random recovery rates. The proposed model is an extension of the traditional t-copula model for the credit portfolio with constant recovery rates. A skew-normal copula model is adopted to represent dependent random recovery rates. In our proposed model, various types of dependency between the defaults and their recovery rates are possible, including an inverse relation. We also propose a conditional Monte Carlo simulation algorithm for estimating the probability of a large loss in the model, and an importance sampling version of it. We show that the proposed Monte Carlo simulation algorithm is relatively efficient compared with the plain Monte Carlo simulation. Numerical results are presented to show the performance and efficiency of the algorithms.