My friend you certainly don't need a degree to become a trader. In fact many successful traders of times gone past have come from all walks of life at varying levels of education.
To be successful getting the correct mindset is the most important asset. No degree will help you in the markets as your mind plays games with itself. No amount of learning theory or technical analysis will help you either, the only thing that will is by learning first hand by getting your hands dirty in the markets.
My advice? Trade for yourself. Why trade for someone else? Learn the basics of technical analysis (babypips.com free school), read trading in the zone by mark douglas, get a solid price action strategy like the pipdaq setups and then practice, practice, practice. Forget a degree, pHD or any education - it will not help you with your goal.
If you're good at what you do, i.e you've taught yourself how to trade which means you have a good overall grasp of the business), and you can convey that to a prospective employer, then I don't think you'll have any trouble getting a job in the industry. Trust me, it all looks far scarier from the outside than the inside and all you need is a simple foot in the door.
I've been employed in several different areas and I'll tell you matter of fact: traders know more about the markets than all the various industry hangers-on, be they reporters, sales-people, managers etc. As long as you can prove you know your stuff, you'll be a shoe-in for any job you go for.
And besides it takes more years to teach yourself how to trade properly than it does to get a degree.
If you mean getting a job in bank on a flow/prop desk then I'd say that your chances are 0% without a degree. Having a degree from a decent university is the bare minimum you need just to apply, to actually get a job you'll need a lot more. It's a sad fact but the competition for these roles is fierce and they have to whittle down deluge of applicants somehow.
Well, get a Job? What do you mean? If just trading forex for your own, it is one thing. But if are trying to become a professional trader for financial institution, sorry, but I wouldn't hire a person with no education, even if you show me Forex account with 1000% of the yearly income. The risks are higher there and the basics do not work anymore.