Qualifications
Here's the good news: you can do lots of the stuff professional paleontologists do without a college degree and years of experience. You can dig and chip away at rocks to your heart's content—assuming you're not trespassing, of course. You can take notes on your findings and hypothesize yourself silly about the time period that likely produced the fossils you found.
What qualities will help you pull this off? First, you'll need a pretty active imagination to think like the dinos did, plus plenty of patience to keep hunting for fossils that don't tend to materialize on demand.
If you want to actually make cash fossil hunting, though, you'll have to get an education first (source). However, most universities don't actually offer paleontology degrees. You'll need a specific knowledge base in math and science to cut the mustard as a paleontologist; that means chemistry, physics, biology, and geology on the science end, and your math skills should go up through calculus and beyond.
You'll also need top-notch computer and data analysis skills, along with digital mapping expertise and a familiarity with satellite-based Geographic Information Systems—like a GPS but way more James Bond-y. Throw in a few communications courses, since you'll write reports and work with other professionals.
So then you're ready to get out and find your own personal T-Rex, right? Not if you're really serious about this whole thing. After all that earlier stuff, it's on to graduate school, where the fun actually starts.
You'll have to do your own independent research projects, for which you'll be accountable to a faculty adviser who probably won't cut you any slack if you try to cut corners. If you perform some valuable original research, you might find a master's or doctoral degree at the end of the excruciating slog.
After all that, you can start collecting all of those bones. Thanks for waiting.