First of all, this feature will only cover the writing of game reviews. This is the area most accessible to newcomers because previews generally require industry contacts (which most novice writers don't have) and only truly established writers get a crack at writing feature-length articles. A second feature will cover managing your relationship with editors and the rest of the games industry.
Making Contact
To answer the initial questions posed above, no one really cares if you went to Harvard, or if you have a degree in journalism. Prior publication, however, can be extremely valuable in separating yourself from the herd. If at all possible, get involved in a fan site and get some writing experience. Don't think writing for a fan site makes a difference? Then consider the fact that GameSpy's very own Warrior got his professional start in games journalism writing Total Annihilation level reviews for me at PC Gamer. I read some of his reviews on a TA-related site, liked his writing, and hired him. And Warrior isn't the only one to go that route.

Star Trek: Armada -- 73%
When first making contact with an editor send a short note introducing yourself while mentioning any relevant information -- prior writing experience, your knowledge of games and the industry, etc. -- and ask for permission to send writing samples. That's it. The whole letter should only be a handful of sentences and should be professional in both tone and use of grammar.
After sending the letter, expect a long wait. Finding new freelancers is not at the top of most editors to-do lists, so it could be a few weeks or even months before you hear anything back. If a month or two passes, you can follow up with another short, professional e-mail. Do not bombard the editor with countless e-mails and never call them. (Let me emphasize that point -- never call an editor.)
After a couple polite e-mails, let's say the editor writes back and tells you they want a 700 word sample review of Diablo II by next Thursday. If the editor specifies a word count and a deadline for the sample review, follow them to the letter. A word count of 700 words means 700 words, end of story. Your ability to meet an assigned word count and deadline are being judged just as thoroughly as your ability to write a review. Make sure you request a set of the publication's guidelines.
Next: Getting down the business...