What does "Megyn Kelly promoted and teased the interview 'within an inch of its life'” mean?

"Within an inch of his/her/its life" is an idiom which means "almost to the point of death" or, more generally, "to an excessive degree".

It is commonly used after the verb "beat", as in "The dog was beaten within an inch of its life."

The writer seems to mean that Kelly promoted the show to an excessive degree.

The verb tease, as used in this instance, is related to the word teaser, meaning (according to Merriam-Webster's Eleventh Collegiate Dictionary)

an advertising or promotional device intended to arouse interest or curiosity esp. in something to follow

So the story is saying that Kelly was using various media interactions to drum up interest in the TV interview she was doing with Trump. As Silenus's answer indicates, describing her as doing such promotions "to within an inch of its [the interview's] life" is an exaggerated and idiomatic way of saying that she was publicizing the interview to the verge of overpromotion—a theoretical state in which media saturation with teasers about an impending event is so excessive that members of the intended audience actually become sick of it and lose interest in it.