We consider the problem of identification of a partially observed finite-state Markov chain, based on observations in a finite set. We first investigate the asymptotic behaviour of the maximum likelihood estimate (MLE) for the transition probabilities, as the number of observations increases to infinity. In particular, we exhibit the associated contrast function, and discuss consistency issues. Based on this expression, we design a recursive identification algorithm, which converges to the set of local minima of the contrast function.