Critical reading is a very useful skill for most PhD students (and postdocs and researchers in general). Instead of assuming that everything you read in a scientific paper is right, it's useful to learn how to evaluate the paper critically: e.g., to question its content, to identify shortcomings and limitations and ways it could be improved.
Experience with reviewing papers is a powerful way to gain experience at critical reading of papers. Writing reviews for journals and/or conference helps a PhD student learn to get better at reading a paper with a critical perspective.
Unfortunately, review opportunities for PhD students are rare.
For instance, it is rare for PhD students to be invited to serve on program committees or asked to review papers. In addition, some may think that PhD students are not competent enough to write correct reviews of scientific papers.
How does one solve this problem? How can a PhD student get opportunities to practice reviewing papers? What would you suggest to a PhD student who wants to do some reviews?
I disagree with the statement that a PhD student never doubt a published article (at least in my field).
This said, it seems to me that one of the first work of a PhD student is to read and "review" papers: for example, when we start a new project, my advisors always ask me to do a whole bibliographical work, sum up the papers read to them, comment them, try to find what is good and what can be improved in the previous work. Even after, I am also asked to always follow on the new papers that could correspond to our work.
I believe it is one of the work of the advisor to help her/his PhD student to learn to do this sort of work.
Then if one really wants to review unreviewed papers, there is always ArXiv (or other equivalent repository for papers) where one can subscribe to the rss feed, then work on reviewing for oneself (or for one's advisor) the papers read that are close to your work.
Since your question seemed to be how can we do a review, I believe it is not important whether it is an official review or not: the important part was to review a paper in the first place.