Within every person lies the Hind-brain (the oldest and most primitive part of the brain). It is this part of the brain where mankind's survival instincts lie. Therefore, this part of the brain is the part which has been referred to as the animalistic part of the human, seen by this brain's other name (Reptilian Brain).
This is important given Shakespeare, while he may not have known of the Reptilian Brain, knew that humans sometimes reverted to pure survival; this is otherwise known as "animal nature."
Therefore, the statement that "Hamlet is about a man's attempt to rise above his animal nature" can be deemed true.
Hamlet must make certain decisions about his rage and revenge. Based upon this, he is a man (a rational and thinking being) who tries to rise above his animal nature (the behaviors which are immediate, unprocessed, and instinctual).
If writing an essay upon this topic, each body paragraph needs to address a specific time where Hamlet chose to be a rational being and not a animal (where the behavior seen would be instinctual).