We present stochastic approximation algorithms for computing the locally optimal policy of a constrained average cost finite state Markov Decision process. Because the optimal control strategy is known to be a randomized policy, we consider here a parameterization of the action probabilities to establish the optimization problem. The stochastic approximation algorithms require computation of the gradient of the cost function with respect to the parameter that characterizes the randomized policy. This is computed by novel simulation based gradient estimation schemes involving weak derivatives. Similar to neuro-dynamic programming algorithms (e.g. Q-learning or Temporal Difference methods), the algorithms proposed in this paper are simulation based and do not require explicit knowledge of the underlying parameters such as transition probabilities. However, unlike neuro-dynamic programming methods, the algorithms proposed here can handle constraints and time varying parameters. Numerical examples are given to illustrate the performance of the algorithms.