"The curse that failed to kill you seems to have forged some kind of connection between you and the Dark Lord. The evidence suggests that at times, when your mind is most relaxed and vulnerable - when you are asleep, for instance - you are sharing the Dark Lord's thoughts and emotions. The Headmaster thinks it inadvisable for this to continue. He wishes me to teach you how to close your mind to the Dark Lord"
-- Severus Snape OP24.
Because of Harry’s vision before Christmas, (involving Mr Weasley being attacked by Nagini) Dumbledore decides that Harry should start taking Occlumency lessons with Snape so that he will be able to close his mind against the Dark Lord.
The sessions take place in Snapes office once a week, and Snape tells Harry not to tell anyone that he is having these lessons. He says to tell anyone that asks what he is doing during these evenings that he is taking remedial Potions.
Dumbledore later says that he was wrong to think that Snape might have been able to overcome the feelings he had about James in order to be able to effectively teach Harry Occlumency. (OP37)
This begs the question as to whether he really ever thought that Harry would be able to learn Occlumency. Considering Harry and Snape's very open mutual dislike of one another, it's seems unlikely that he would of thought that Harry would be able to master Occlumency, especially when you consider that one of the most important factors is being able to rid your mind of all emotion. Dumbledore is also very aware of Harry's suspicions that Snape hasn't really come over to their side, so all in all, it seems these lessons were doomed from the beginning.