Network in the news office. If you are in the newspaper office, ask around for contacts at other publications. See if your office could use a fact checker even part-time. Try to get your foot in the door by filling a similar role like proofing or co-authoring stories.
Get a specialized degree. A communications degree from a highly rated school with a concentration in research will make you among the most qualified for fact checking jobs. A few classes on a booming field like technical writing wouldn't hurt either.
Leverage your contacts and name recognition. Try to find your way up or across to someone who deals with fact checkers within the industry. Putting together a portfolio of your work will help you get noticed by those who could have the authority to get you into an advanced role like fact checking.
Go corporate. A big, corporate shop is most likely to have a fact checking job available, and with a foot in the door, you'll be one step closer to getting there. Types of corporations that are promising include mixed-media companies and companies with television or mass media holdings. Big pharmaceutical or financial shops are other venues: their need for trade publications could pay off for you.
Contact editors with freelance staff. A swamped editor might consent to hiring on someone you could call a "fact checker", a kind of executive assistant whose job would be to follow up on stories from the field to ensure quality. This can be a long shot, but it's one way to pursue your niche job in the greater picture.