How do you correctly place possessive apostrophes when there is more than one noun?

The class of Mr. Wood at Washington High School would be best expressed as 'Mr. Wood's class at Washington High...' or 'At Washington High, Mr. Wood's class...'

Shortening the term to 'Washington's Wood's class' assumes that the listener knows that 'Wood' is a person and that 'Washington' is a school, which may not be the case. That form would only be appropriate in casual conversation with people who were familiar with these facts.