Odds of Getting In
Some people start out their public lives as diplomats and then move up in the ranks or sort of switch focus. A good example is Thomas Jefferson, who was a diplomat to France before he became the third President of the U.S. And some people (TV and radio personality Art Linklater became ambassador to Australia and Chilean poet Pablo Neruda served as ambassador to France) are appointed by the President and confirmed by the senate after having led successful lives in a variety of areas.
That being said, as most diplomats are Foreign Service Officers, in order to become one, you have to jump through many hoops (See Qualifications) and the competition is fierce. Apparently, an extremely selective few get in: The FSO is more selective than Harvard.
If you're sure this is what you want to do, the Foreign Service exam is given once a year in an area near you. Visit the U.S. Department of State website to find out more about how to apply for a job as a diplomat.