The process to become a website editor can be different for everyone, depending on the website you wish to work for. Some websites are simply online magazines or similar types of media businesses, and are structured very much like any other company in which writers and editors need to work their way up. Another option is to simply start your own website if you feel you have a good enough idea, and become a website editor that way. The beauty of the Internet is that anyone can start a website, either for free, or for a very low cost.
A website editor is generally similar to the editor-in-chief or managing editor at a magazine, though depending on the size of the website, and the different topics it covers, it may also have departmental or subject editors as well. Anyone wanting to become a website editor should be familiar with the structure of the website and content of the material included there. Some experience editing for the web will almost certainly be required; this is because online writing is becoming increasingly focused on search engine optimization, and editors of a website will really have to be familiar with these tools.
If you want to become a website editor you might start out doing tasks such as copy editing or fact checking for other websites, or even the website that you want to work for. This helps you develop important web editing skills, and increases your marketable experience. In addition, if you already work for your desired employer, you are much more likely to get promoted into an editorial role than to get hired into one from the outside. It is important to always do your work well, and to develop a good rapport with your superiors. A degree in English, journalism, or communications certainly can't hurt either.
Of course, another way to become a website editor is just to start your own website. You can literally start a website on any topic you choose. Purchasing a domain name and hosting plan is not too expensive, or there are many options for free web hosting if you wish to save money. You may wish to visit some other successful websites you enjoy reading in order to get ideas for what works well and what doesn't; of course, be sure never to directly copy or plagiarize anyone else's ideas when you are getting your own inspiration.