There are currently 12 actors who have played the Doctor, and this also fits the title of "when am I?" referring to the Doctor being a Time Lord. Time Lords also have 2 hearts which would make sense with "Two-beat".
My first was a grandfather,
The First Doctor, William Hartnell, was known as "grandfather" by Lady Larn.
My second said when to run,
The Second Doctor, Patrick Troughton, was well known for the phrase "When I say run, run!"
My third was then a scarecrow,
The Third Doctor, Jon Pertwee, played Worzel Gummidge (a scarecrow) in the 1979-1981 series Worzel Gummidge.
My fourth was there for longest,
The Fourth Doctor, Tom Baker, was the Doctor for the longest (around 7 years).
My fifth was a father-in-law,
The Fifth Doctor, Peter Davison, had a daughter, Georgia Moffett, who married David Tennant, making him the father-in-law of the tenth Doctor
My sixth was like my fourth,
The Sixth Doctor, Colin Baker, shares the same last name with the fourth Doctor
My seventh changed a bit,
The Seventh Doctor, Sylvester McCoy, started as a comical character, but later developed a darker nature.
My eighth came only once,
The Eighth Doctor, Paul McGann, only appeared as the Doctor once
My ninth was once obscure,
The Ninth Doctor, Christopher Eccleston, also played the role of Jude in the 1996 film of the same name, based on the novel Jude the Obscure
My tenth was no landlord,
The Tenth Doctor, David Tennant, was "no landlord", he was a Tennant (ba dum tsss)
My eleventh was married,
The Eleventh Doctor, Matt Smith, marries River during the episode The Wedding of River Song
And my twelfth is an idiot.
The Twelfth Doctor, Peter Capaldi. This is referring to the quote, "You know who I am? I... am... an idiot! With a box and a screwdriver."