Choose your topic. A good essay will have a clear and narrow focus, so don't try and take on too much at once.
Find out all the information you can on your topic. It sounds obvious, but if you start writing before you really have something to write about, you will soon hit a brick wall. Try to find the most up-to-date information available, as this will increase your chances of saying something original and engaging.
Write your thesis in one or two sentences. The thesis is your main argument, presented in a way that tells the reader your view of the topic you're discussing. Your argument has to be provable--or at least persuasive--given the information that your research has provided. A simple example of a thesis written for an essay about the importance of education might be, "Teenagers whose parents value education are more likely to attend college."
Plan your essay. An effective claim argument plan should show, in separate but logically connected paragraphs, how the argument progresses from start to finish. Use subheadings to help clarify the different sections, if desired.
Construct the first paragraph. Introduce your topic so the reader knows exactly what he is reading about and what claim you are going to make. Insert the thesis sentence(s) you developed earlier into this first paragraph. Writing short and simple sentences is the best way to begin to get your points across.
Write your way through the essay, taking it one paragraph at a time. Use words such as "therefore", "consequently" and "however" to tell the reader how the different sections of your argument build on each other. As Bryan Greetham notes, "To create a taut, cohesive piece of work, each paragraph has to have a clear connection with the one that preceded it."
Conclude your essay by giving a retrospective overview. Don't be afraid to tell the reader what she has just read. Phrases like "as we have seen" and "as I have shown" are useful here, and if you conclude with slightly different words and phrases than you used in introduction, your argument will be clear without being repetitive.
Proofread your essay for grammatical accuracy and the quality of your argument. Read it through slowly, and in each section, ask yourself this question: "Does this develop my argument in a persuasive way?" If you can get from start to finish, answering "yes" for each paragraph, then your claim argument essay is done.