The process of publishing a popular science article is different depending on what has led up to the article itself.
If you are a researcher and got contacted by a journalist, the process will most likely run itself. In this case you should always as the journalist to look over the finalized article before it is published. Not so much for language and grammar, but partially to ensure that there was no misunderstandings during the interview and partially to see that you have been quoted correctly.
If you are communicating your own research or write about a scientific topic, you must figure out how to make your article be seen. If you publish on a blog, make sure to have an associated Twitter-account, where you can advertise your contribution. Use hashtags like #scicomm and #science, and investigate if there is a specific hashtag used by journalists in your country, who can pick up your article and share it in their own network.
Another approach is to contact the science magazines and news papers yourself. Ask if they are interested in publishing your work. Some medias pay the author, others do not. The best way is to go ahead and ask.The websites of the magazines usually have a contact form or mail to contact. The information might be well hidden (because the majority of visitors are there for the content and not the contact information of the editors), but it should appear on the website.