The core verb phrase here is "to expect something of someone". In this case, the underlying subject is the teachers, the 'someone' is the students, while the 'something' is the expected standard of behaviour and achievement.
This phrase is usually used in the passive voice, as it is in your example. Here's your deconstruction:
What = subject (interrogative pronoun) ---> refers to the students' duties
is expected = verb (passive voice)
of all of you = indirect object --> refers to the students
A paraphrase, in the active voice, would be something along the lines of "What do we expect and require you to do this school year?".
This is a common way to talk about what someone's duties are in the eyes of others. For example, "What's expected of me?" means "What do other people believe that I ought to do?/What do other people believe that I'm capable of doing?".