If I write a story about an alien time traveller from a dying race, with encyclopedic knowledge, who sporadically visits London, travels with young companions, occasionally saves Earth and other planets, and appears to change bodies periodically. Would this be a breach of copyright?
There is too little detail in your question to give a clear answer. In fact, to get an authoritative answer, you need to let a IP lawyer read the full story and give a legal opinion.
The answer depends on how you write the story.
Copyright gives the copyright holder control over derivative works, so if your story can be classified as a derivative work of Dr. Who, and is not covered by the exceptions that exist in copyright law (i.e. Parody, Fair use), then you would breach copyright.
However, if your story just happens to use a similar story framework as Dr. Who, but where the plotlines, settings and and personality of main characters are all different from Dr. Who, then it probably will not be considered a derivative work and you would be OK.