Unfortunately the BBC "have there professional writers," I myself looked this up to do, but they "send back all letters unopened" :/
Not many large companies will accept unsolicited stories from unknown writers. First, to be blunt most aren't very good and even if the idea has merit, would require a lot of work to make it to TV. Second, a lot of people might have come up with your "original idea". That's why they won't even open and read them. If it bears any similarity to something they already have in the works, it can open them to a lawsuit.
The only way to do it is to work your way up. Get yourself published. Make contacts. Understand that there are probably a ton of people who'd love to write for any television show, let alone Doctor Who specifically.