Get an internship at a literary journal. Internships are a great way to get a foot through the door. They provide the opportunity to not only make contacts and gain experience, but to learn the ins-and-outs of the literary journal business. Interns generally begin by doing basic office work and reading submissions.
Make contacts. Contacts will be helpful either in getting a job in a literary journal or in getting references for future job opportunities.
Apply for a job as associate editor. An associate editor is someone who reads submissions and determines whether they should be sent to the managing editor. An associate editor is usually an entry-level position at literary journals. Research which journals are hiring and apply for them.
Learn everything about editing. Make sure you know the details of your literary journal's house style, which are the grammatical rules, and font choices the magazine uses as a general rule. Be familiar with its formatting and design and help out by working with printing presses, proofreading galleys, and dealing with the budget and/or fundraising. Knowing the ins-and-outs of running the journal will be helpful in getting promoted.
Work toward becoming managing editor. A managing editor oversees many aspects of a literary journal, from working with associate editors and graphic designers to preparing and sending the journal to press. A managing editor is generally one step below the editor or editor-in-chief, so she must know the literary journal inside and out, and she understands how to get it out to the public. Becoming a managing editor, therefore, is the last step to becoming an editor.
Look for opportunities to become editor. Once you have relevant experience and references, you might be a candidate at your own publication and at other literary journals.