He was pretty good, no? Picture via Don3rdSE. Usual rules apply.
Advertising's not your typical career. It's not something where you need to spend a few years converting your degree and winning a training contract, like Law. It's not like Dentistry, which takes seven years of study, with many different specialisms. It's not even like becoming an Actuary, which requires a degree, and then passing many, many exams over the next five years or so (hell, it may be even more than that).
In fact, you don't need a degree at all. And, God knows, with degrees not being the mark of the academic elite, it may even be better to get a bit more life experience, or try your hand at something entirely different - quite a few planners I know are wannabe rock stars.
I'd go so far as to say it doesn't matter your exact background, as all the people who succeed in advertising are gifted amateurs. It's not to put them down, it's more to say there's no precise formula for getting in and on. You just have to persevere, and use your instincts.
Obviously, there are things you can do to improve your chances - such as looking and tailoring what you offer to the agency you're applying, so they can't easily turn you down. Some are more trendy and appreciate extra-curricular bits and bobs; others like some proof of organisational skills. Either way, both are really after gifted amateurs.
You have to be gifted, in that you have to want to understand people to discover things and do well. This is not something that'll happen tomorrow or by next Tuesday. It's something which requires you to put your head down and watch and learn, whether you're an account handler, planner or a creative. It's this which leads to the most interesting lateral thoughts; seeing what everyone has seen and thinking what no-one else has thought (yes, i'm ripping off Albert Szent-Gyorgyi).
You'll never really know for sure that you're gifted; it only happens at the end, when you can look back on a career founded on perseverence, thoughtfulness and a bloody minded devotion to doing the job to the best of your ability.
Also, the amateur part is equally important. If you try to enter the business as a po-faced, incredibly serious 'professional', it becomes difficult to do well. First of all, people will wonder what led to that rod being inserted up your arse. Far better to acknowledge that you DON'T always have all of the answers. Why can't you admit you just have an intelligent opinion based on your own knowledge and research, rather than assuming you know it all?
This helps stop smug planning, over officious account management and junior creatives who make a TV ad and assume they've inherited the combined skills of Bacon, Dali and Shakespeare.
If you have an instinct for people and care about your work (that means taking care of your own output, NOT on getting qualifications which you can point to and say you're an 'advertising professional'), you'll do well.
In short - don't assume you've got all the answers. Do believe in yourself, like ol' Cassius, and keep trying. People will like you a lot more, and you'll do more interesting work.