Odds of Getting In
There are two ways to go about this: with another company or on your own. Going the company route means distribution agencies, sales offices, and studio acquisitions. Each of these options has their perks and drawbacks, but typically you'll need some combination of great references, a bachelors' degree, and/or an A-type personality.
Two out of three should at least get you in the door, although those studio gigs really want you to have a track record. Even the Warner Bros. head honcho had to get his start somewhere―Six Flags to be precise (seriously, this guy is our new hero).
If, on the other hand, you want to branch out on your own, there are opportunities for this, much in the same way there are opportunities to become a five-star chef by working at a fast-food chain. If this is how you choose (or are forced) to begin, you either have to have some major family connections to pull from (in which case, you'll be fine) or you have to get out there and start meeting people.
This means you'll be doing all the stuff you'd be doing at a studio or distribution agency, just on a larger, more desperate scale. Every film fest, every workshop, and every Meetup, Tweetup, and Instaparty will offer the chance to walk through the door to your future.