By definition, a Markov chain only depends of the current state. That said, I can think of a way to make something depending on the past as well: if you have two states (S for sunny, R for Rainy), with two days of state you now have four states: SS, SR, RS, RR, which gives you a transition probabilities matrix - of course you will have some restriction, for example from SS you can only go to SS and SR.