The next step is to present your claims and support them with relevant evidence. Make sure you've thought everything through and have enough information to cover all of the points of your argumentation. Each major claim you mention should have from 1 to 3 supporting facts or testimonials. The only way to convince a person with an opposing view is to present the undeniable facts. The best way to defeat your enemy is to use his weapon against you. The same applies in the case of academic writing as well. You can take an opposing argument and refute it. It will show your readers that you have a profound knowledge on the topic and are ready to defend your point of view no matter what.
The function of an argumentative essay is to show that your assertion (opinion, theory, hypothesis) about some phenomenon or phenomena is correct or more truthful than others'. The art of argumentation is not an easy skill to acquire. Many people might think that if one simply has an opinion, one can argue it successfully, and these folks are always surprised when others don't agree with them because their logic seems so correct. Argumentative writing is the act of forming reasons, making inductions, drawing conclusions, and applying them to the case in discussion; the operation of inferring propositions, not known or admitted as true, from facts or principles known, admitted, or proved to be true. It clearly explains the process of your reasoning from the known or assumed to the unknown. Without doing this you do not have an argument, you have only an assertion, an essay that is just your unsubstantiated opinion.