In what ways did the tragic deaths of Brutus and Caesar result from their nature or character?How would I begin an essay introduction with the answer?
Very good question. I will sketch a few ideas for you to develop and then consider your question about the introduction to help you frame your response.
To answer this question you need to go back and consider the characteristics of these characters and how they are exhibited in the play. In particular, you need to focus on their shortcomings or characteristics which are exploited by other characters and result in their downfall.
For example, Caesar´s major fault that results in his assassination is his own arrogance or sense that he cannot be harmed. As Calphurnia says to him in Act II scene 2, when she is desperately trying to persuade her husband not to leave that day:
Alas, my lord,
Your wisdom is consumed in confidence.
It is this sense of his own strength and invincibility that leads him to ignore the warnings of his wife, the Soothsayer and others:
Caesar shall forth. The things that threatened me
Neér looked by on my back. When they shall see
The face of Caesar, they are vanished.
Such a strong self-belief in his own strength and power is arguably what leads to Caesar ignoring the premonitions of other characters and his assassination.
With Brutus we need to consider how his honour, his defining characteristic, is manipulated and played upon by Cassius in the famous seduction scene in Act I scene 2, and then leads him to make some misguided decisions. Chief among these is his underestimation of the character of Marc Antony. It is Brutus who persuades the conspirators not to kill him and then allows Marc Antony to address the crowd. He dismisses the character of Marc Antony as one how is "given/to sports, to wildness, and to much company." He also wants to strike a moderate balance between the murder that he sees as necessary and an all-out slaughter:
Our course will sem to bloody, Caius Cassius,
To cut the head off and then hack the limbs,
Like wrath in death and envy afterwards;
For Antony is but a limb of Caesar.
Let´s be sacrificers, but not butchers, Caius.
In fact, Brutus´ own sense of honour and what we can identify as his naivety leads him to ignore the advice of the far more politically aware and shrewd Cassius.
In your introduction then you need to introduce your two key ideas about these characters and briefly explain how you are going to elaborate on their characteristics. Remember, in the introduction you are to sketch an outline of how you are going to answer the question. The biggest failing of most students is that they begin to answer the question in their introduction - this should only be done in the main essay. Good luck!