Read the research paper straight through without marking a word. Pay attention to the content of the paper. Keep in mind the author's thesis and take special note as to whether the author's supporting reasons and evidence stays focused on the thesis.
Go back to the introduction. According to authors Wayne Booth, Gregory Colomb and Joseph Williams, the introduction should contain a problem and the writer's solution. The solution is the thesis. The thesis is the most important part of the paper. It's vital you understand the significance of the problem and the author's claim (or thesis) for solving it. If you do not, write down how the author can make this more clear. If everything is understandable, compliment the author's clarity.
Read the body of the research paper paragraph by paragraph. Examine each of the author's reasons. Does her reasons support her thesis? Does she provide enough evidence to show how and why the reasons support the thesis? Mark passages you feel provide weak support. Compliment the author where her reasoning is strong.
Read the conclusion. The conclusion should restate the research paper's main points and thesis. With many research papers, the author also will discuss the implications of his findings and whether more research needs to be done. Comment on whether you feel the conclusion ties together the paper or ends weakly.
Check grammar and spelling. If you have not already, look for typos and mistakes. If you're not comfortable with correcting another person's grammar, then don't. It's far better to not correct an author's mistake than to introduce one into her paper.
Look at the citations. Does the author ever neglect to mention where he found a piece of research? Or not cite a direct quote? Also check to see if he conforms to a specific style (for example, MLA or APA).
Offer an overall critique at the end of the paper. While you should have been writing your notes alongside the paper's margins, write an overall assessment at the end of the paper. Discuss any persistent weaknesses or strengths. One recommended style is to give a strength, a weakness and another strength.