One thing, fairly specific, that might be interesting to write about are the characters of Iago and Hamlet as they share this similarity: confiding to the the audience their real feelings about the events of the play.
It was a common dramatic convention of Shakespeare's day for characters alone onstage to speak directly with the audience in soliloquies. Actors in Shakespeare's theatre never pretended that there was an imaginary "fourth wall" that separated the action onstage from the audience. So, actors (in the guise of whatever character they were playing) often spoke directly to the audience, including them in the action.
Both Iago and Hamlet have just these sorts of moments of soliloquy alone onstage with the audience. Both have schemes they are sharing with the audience, and both are also pretending to be something that they're not -- a fact each shares with the audience. While Hamlet is pretending to be "mad," Iago is pretending to have Othello's best interests at heart.
Comparing these two charcters' interactions with the audience in their soliloquies might be a fun topic.
Good luck with your assignment!
If you have a particular assignment task or description, I would consult that first. The next step would be to discuss this with your teacher or instructor to see if there are any elements that you need to integrate in your writing that might help you answer what to write about regarding both plays. There are a variety of topics that can be written and discussed in a writing sample comparing both works. One immediate location would be to discuss the main characters in both dramas. How are the protagonists similar to one another and how are they divergent from one another? What traits do they hold in common and how do their respective tragic conditions reflect their fates? There is much in terms of both characters and how they are depicted in the course of their developments. Another element that can be discussed is how both dramas show the plight of women. Desdemona and Gertrude or Ophelia could be a very interesting writing sample.
I have always found that trying to puzzle out Iago's motivation for his actions in Othello is a fun essay. Shakespeare gives us hints, but never really comes out and states a concrete reason. It leaves us wondering if Iago is just simply evil, or if he is seeking vengeance for some past wrong that we do not entirely understand. If you wanted to compare and contrast Iago and Hamlet in terms of their motivations, that could be a very good essay also. Hamlet can be a deeply contemplative character. He considers his actions carefully and we get to see glimpses of his internal conflict over the mission his father has charged him undertake.
You might end up with the proposition that Iago and Hamlet are the anathema of one another. You might decide that Iago has no motivations other than an amorphous desire to hurt those around him, while Hamlet has too many motivations and they slowly tear him apart (He wants to kill his uncle, he hates his mother for wedding his uncle, the ghost charges him not to harm his mother but to kill his uncle, he wants to prove he is better than his uncle who killed the king before he could pray, yet when he has a chance to kill his uncle after his prayers, he can't do it because he thinks this will permit the murderer to go to heaven, etc., etc.)
On the other hand, you might conclude that Hamlet's motivations are no better than Iago's. Is Hamlet merely seeking revenge for the death of his father? If you decide that Iago is not simply evil, but is seeking vengeance, is his revenge (for being passed over for promotion and/or cuckolded) any less justified? You could go many different directions here, and these plays will give you so much support for whatever answer you decide on that you can't be wrong.