This is a good assignment that is being used to test your knowledge of the play but it is also after your own opinion and thoughts and it is testing your ability to express those in a letter form. So, the best place to start is for you to think about what you thought was funny or particularly effective about the methods that Petruchio adopted to "tame" his wife. When you were reading or watching the play, what amused you? I would say that you probably need about three different examples from the play, each of which that can form a middle paragraph of your essay. For each paragraph you need to offer a brief summary of what Petruchio did and then say briefly why you think it was effective. Consider the following example.
I loved the way that, in Act IV scene 1, Grumio relates to Curtis how when Katharina's horse fell and she ended up in mud, Petruchio did nothing to help her but only beat Grumio so that Katharina had to wade through the mud to pull him off Grumio. That was an excellent way of humbling Katharina and also to stop her thinking so much about her own needs and desires. Beating Grumio forced her to think of others for a change.
This is just a quick example, but I hope you get the idea. Good luck!