There is no set career path into the business. Many of my contemporaries started out in television or radio doing other jobs, and wrote music in their spare time. Others started out life as artists and producers in the record business, before later moving to television writing, and again there are others who were classically trained in music. I began my career in local radio, and moved on to television as a location sound recordist. I spent ten years or so in this job, writing music in my spare time, before I had enough music work coming in to be able to do it full time.
I would recommend that you start by writing music just for fun, and have another skill to fall back on if it doesn't work out.