How do you write an opening and closing paragraph for an essay comparing Macbeth and Richard Nixon in leading themselves to their downfall?

Firstly, you need to decide what the central points of comparison between Macbeth and Nixon are. While you will go into depth in the body of your essay, you need to give your audience a snapshot of what you plan on saying later—and all of that goes in the opening paragraph.
In the opening paragraph for this particular assignment, you might want to begin with the theme of the ambition and unscrupulousness of politicians being a constant theme in both literature and history. This would show how real life and art often intersect, which would be a good segue into comparing Macbeth and Nixon. Briefly outline the similarities (or even differences) between them, but try not to make your introduction too long: 5–7 sentences usually does the trick.
And of course, you'll need a thesis statement at the end of your opening. A thesis statement is a single sentence which sums up your basic argument. Make sure it's not too broad ("Macbeth and Nixon are very similar as politicians because both were responsible for their downfalls"); try to be as specific as possible by pointing out, perhaps, shared qualities that lead to their downfall or perhaps how different elements lead to them having similar downfalls.
Nailing the closing paragraph can be a bit tricky. Unlike what some will tell you, do NOT just reword your opening and slap it onto the end of your essay. Think instead of the closing as the last chance to emphasize your points to the reader. For this particular assignment, perhaps you could say that Macbeth remains relevant because even into the modern era, we continue to see corrupt political practices.
And most importantly regarding conclusions: do not bring up any new points or arguments there. The conclusion is where you wind things down. If you come up with any new insights while writing your essay, then give those insights a body paragraph. They do not belong in the ending paragraph.