Qualifications
Many copy editors have a degree in journalism, but most have several years of experience working as a writer (source). People make the transition from writer to copy editor by working as an editorial assistant.
Editorial assistants perform regular administrative duties on top of going through unsolicited queries from writers, acting as a liaison between the editorial department and writer, maintaining files on manuscripts, and attending editorial meetings. However, being an editorial assistant doesn't always mean that you're a shoo-in for the editor position if your boss ever leaves the publication.
Much like any job, editorial assistants must actively volunteer for more responsibilities.
"Need a clever headline written? I can stay late to proofread."
Also, more publications are using software programs like Quark. Quark is a publishing platform that publications use for layouts. Publishing companies also use other types of software such as Adobe InDesign. Educate yourself in whatever software your editor or design department uses for the layout.
There have been cases during the last couple of years of companies delegating design and editing duties to one individual. Again, like any other job, knowing too much is not going to hurt you. Well, as long as you don't work for a bank buying up bad mortgages.