‘I think it’s a sad, sad thing that we tend to remember bad things more clearly than we do good things. But actually that’s quite helpful in this case,’ laughs Alan Cumming. The actor and entertainer has just released his new book, Not My Father’s Son but it’s not your average celebrity autobiography. Rather, it’s an emotional ‘family memoir’ and is as much about his mother, brother and, crucially, his physically abusive father as it is about the star.

‘When someone famous writes a book about themselves, you think it’s going to be “I was born and blah blah blah and my first box office success blah blah blah”,’ he jokes, speaking to us from the US while promoting the book Stateside. ‘The book is called a memoir in America, but Canongate [his UK publisher] put in the “family” word. I really like that and wish they’d done it here. It confounds your expectations of a celebrity autobiography: I like that. I think that me being famous is a little bit of the story but actually the bulk of it is not connected to that.’

To his fans on either side of the Atlantic (he divides his time between New York and Edinburgh), Cumming has recently become best known for his role as ruthless politico Eli Gold on The Good Wife. But this former RSAMD-graduate began his career at local theatres like the Citz and the Lyceum, before cracking the West End as the Master of Ceremonies in Sam Mendes’ 1993 production of Cabaret and springboarding into a role in 1995 film Circle of Friends.

But there’s little of this career detail in Not My Father’s Son, which switches between past tales of Cumming’s childhood in 1970s Perthshire and his life in 2010, when he began filming an episode of the BBC’s genealogical history programme, Who Do You Think You Are?. Hinging on his father’s disturbingly violent treatment of his sons, it’s deeply emotional stuff yet completely surreal in places. The book opens with his father forcibly cropping young Alan’s hair with a pair of rusty clippers, pressed against a workbench in a shed. In the next chapter, Cumming is shuffling on stage at a Cinema Against AIDS gala at Cannes, alongside Patti Smith and Marion Cotillard, just as an unlikely duet between a bemused Smith and Mary J Blige is being auctioned off.

Peppered with flashbacks to his past – usually violence at the hands of his father, but occasionally heartfelt encounters with his much-beloved mother – the thrust of Cumming’s memoir takes place in 2010. Just as he is about to begin investigating his maternal grandfather’s history with the BBC team, he receives some shocking family news delivered in a panic by his brother the night before filming begins.

Viewers who’ve seen Cumming’s Who Do You Think You Are? episode may remember its bewildering conclusion, and that particular journey is described in engaging detail in Not My Father’s Son. But the book’s focus is the other mystery that was simultaneously unfolding in Cumming’s life, away from the cameras: one that involved confrontations with his estranged father (who passed away from cancer shortly after), DNA-tests and fraught soul-searching.

‘There were times I just thought “why is this happening to me and what the fuck is going on?”’ he says. ‘You know, it was overwhelming. And that’s partly why I wrote the book. I want to go “look, look at this, people, this fucking crazy shit happened to me.” By doing that, you you’re giving it weight, you’re giving it its import. I want to give weight to what happened to me in my childhood, and also what happened to me that summer [in 2010]. It was big and I want to mark it.’

And so far, the critical and public response has been exceptional. ‘It’s been amazing, actually, very overwhelming,’ he admits. ‘I’ve actually done some readings where people have come up to me and said they were very moved and talked about their similar situations. So, it’s been quite an overwhelming time for me, I have to say.’

Although he’s written before (his debut novel Tommy’s Tale came out in 2002), Not My Father’s Son shows Cumming to be a particularly skilled storyteller. His personal drama is spun out in a thrillingly mysterious way, and it’s even won plaudits from lauded thriller writer Harlan Coben. His acting career might fill his time just now (in addition to The Good Wife, he’s revived his iconic Cabaret role on Broadway this year) but he says he hopes to write more in the future.

So might we soon see him join the ranks of Scotland’s world-renowned crime novelists? ‘I don’t want to step on anyone’s toes,’ he chuckles. ‘Actually, I gave it to Ian Rankin to read and he said “gosh, it’s like a thriller” and started joking about stuff. I mean, I do love a thriller. But I think this is a certain example of truth being stranger than fiction, so that was kind of in my favour.’

He may not be joining the Tartan Noir set just yet but in this deeply touching memoir, Cumming has certainly marked himself out as one of our bravest and most perceptive writers, and an unexpected literary talent.

Alan Cumming is at Assembly Roxy, Edinburgh, Wed 12 Nov; Not My Father’s Son is out now published by Canongate.