Qualifications
Well, no one really knows. There are a couple of degree programs for aspiring syndicated cartoonists. They tend to fall into the category of Bachelor of Fine Arts in Illustration (source). However, there aren't very many of these. More to the point, none of the successful cartoonists we're aware of have one of those hanging on their wall. Most seem to have followed unique paths.
For example, Bill Amend, the creator of Foxtrot, is a physics major who just happened to like to draw. We're sure his parents were just thrilled that he took his hard earned (and no doubt expensive) physics degree and used it to draw a comic strip. But it worked out pretty well for him. And his educational background made his strip unique. It often included science, math, computer, and science fiction references that gave it a smart and different flavor.
It should come as a shock to no one that Scott Adams, creator of Dilbert, was once a white-collar cubicle dweller. Eventually, his and his colleagues' work experiences inspired the corporate hopelessness of the world Dilbert shares with Dogbert, Wally, Alice, and the Pointy-Haired Boss. Gary Larson is an anthropology and music enthusiast, and those themes made frequent appearances in The Far Side.
There isn't a specific set of criteria. Take a look at Beetle Bailey, Hagar the Horrible, and Pearls Before Swine, and you'll see that even above-average artistic ability isn't even necessarily a requisite. Nobody else in comics could draw like Watterson, yet many before and many after him enjoyed great success.
As long as you have something worth saying, can say it with at least a little humor, and can make an audience appreciate it through your pictures and words, you have a shot.