When David Morrissey was a teenager, he gave up on school. Not academic, he had discovered acting, and that, as far as he was concerned, was that. "All of us, at some point, find the thing that keeps us ticking. Sometimes it lasts a lifetime and sometimes it lasts a couple of months," he says, with rather more surety than that sentence deserves. "I sat outside the Everyman [theatre in Liverpool] the first time I went, and I could hear what was going on through the door, and I knew that if I went in, my life would change. And it did. It gave me a life, that's what it gave me."

This was Liverpool in the 70s. "The youth theatre had a great creative energy, but the theatre itself … Jonathan Pryce, Bill Nighy, Julie Walters, Antony Sher were there just as I got there. It was all coming at me, this creative force. I never wanted to be anywhere else."

Morrissey is the kind of actor whose name, when you see it on a cast list, makes you feel reassured you're settling down to watch something good. He was astonishing as Gordon Brown in The Deal, the drama that told the story of the leadership agreement made between Brown and Blair. State of Play, in which Morrissey played the compromised MP Stephen Collins, is still one of the best dramas the BBC has ever made, and his role as the corrupt detective in the Red Riding trilogy was one of pure menace. Next year comes a film he made with James McAvoy and Mark Strong, Welcome to the Punch – a London-based crime thriller already being talked about as a step up from generic gangster films.

He doesn't know why he keeps being cast in roles that require the kind of dour inner turmoil or quiet villainy he brings. "You get that box you're put into. I think it's being six foot three and a miserable bastard." Is he that? He's clearly a man who spends a lot of time in his head, but he smiles a lot, too. "No, I'm not really."

Still, more darkness calls. Morrissey joins the third series of hit US zombie drama The Walking Dead as the governor, a beast of a man who runs a small town called Woodbury. For anyone who hasn't caught the first two seasons, based on a comic book series, it's about a group of survivors living in a post-apocalyptic Atlanta where most of the population have become shuffling, shambling zombies, or "walkers".

Zombies have long carried weighty cultural significance. If George Romero's 1968 Night of the Living Dead zombies were referencing growing nuclear fears, and 28 Days Later came out at the height of the war on terror in 2002, what do the Walking Dead's flesh-eaters say about our time? "The unknown, I think," he says. "The unknown threat. The idea of being besieged by an uncontrollable populace is very frightening. The idea that there is a non-negotiable enemy out there. There is a sense of the besieged community and that's an interesting place for me. Woodbury is a community in the heart of a dangerous place that is finding a way of surviving. It has barriers and fences – it lives for its security. What the zombies represent is the instability and slightly crazed enemy."

David Morrissey as the Governor in Season 3 of The Walking Dead

And the governor, then, is an authoritarian who uses brutality and fear to control his community. "He has to play this game with the populace, which is about reminding them how dangerous it is, so they stay within his governorship, and feel grateful to him for what he is providing," says Morrissey. "He has created the world he wants to create in this madness, but as we know, power is a great corrupter."

We meet in an overheated hotel room in London. Morrissey is home for a week. In the past, when US TV jobs have come up, Morrissey has never been able to commit to them – at the pilot stage, it's typical for actors to sign up for seven years – because the timing hadn't been right for his young family (Morrissey and his wife, the novelist Esther Freud, have three children, the oldest 17). Although he has to spend half the year in Atlanta and is signed up for five years, "that's the great thing about the zombie apocalypse – there's a way out, I guess. This seemed the right time. It happened really quickly for me. I phoned my wife and said what do you think, and she said: 'Go for it.'"

His is a career notable for its few missteps, but this means they're more conspicuous. He played the lead in Basic Instinct 2 opposite Sharon Stone, a film so bad that even when his friends bring it up, he says, they do it with the sort of hushed, disapproving tones "like they're saying, 'What about that time you were in the BNP?'" He laughs.

"It was a film that didn't work, get over it." The only other smudge I find on his CV is a voiceover for a McDonald's ad.

"Yeah," he says, sounding pained. "I felt quite conflicted by that. I did it … It was one of those things I wish hadn't happened, but I did it, and I justified it to myself at the time. But yes, it was a blip, I absolutely wear that one."

There has been much talk recently about how only posh kids can afford to become actors now; Julie Walters and Ken Loach have weighed in. "It is sad that so many of the young leading actors are coming from such a narrow social background," Loach said. "It emphasises the fact that this is a society based on class and that privilege confers status."

Morrissey, whose father was a cobbler and whose mother worked for Littlewoods, agrees, but adds that acting is not the only profession closed to many young people. "Certainly in politics – interns are mostly from middle and upper classes, because those wages are going, and I presume that's true in most industries. There's an age bracket – late teens to early 20s – when, if your parents don't have the money, you're not going to be able to do unpaid internships. Acting reflects other industries. You can't survive on those wages at that level, so they have to be supported by parents, and so that has to be the middle-class kids, and that's a disgrace."

He pauses. "I think people are right [to complain], but it's always been hard for those from a working-class background to get into the arts. It's not going to stop people, but it will make it harder."

Morrissey trained at Rada. Would he have gone to drama school if he'd had to pay fees? "No, there was no way I could have done that. But whether that would have stopped me being an actor is a different question. Fees for drama schools are ridiculous, and I know the drama schools feel that as well. At Rada, which I go back to every now and then, I do see a diverse demographic. But they have to struggle to subsidise people; there are bursaries and scholarships. Where the answer lies I'm not sure. Maybe it's the industry itself putting money back in."

It's always difficult, he says, in times of economic crisis, for the arts to champion itself, "when something like the NHS is struggling, but it's still our job to do that". He is supporting the campaign to raise money to fund the rebuilding of the Everyman theatre. Has he seen the effects of the cuts on theatre? "You see younger actors working for a good theatre that gets good audiences, but being asked to work for £100 a week. How are they going to live on that, particularly in London? Touring was very important to me as a kid. There were some really strange experimental theatres – it was wonderful to see how diverse theatre could be – and that's being cut. We're spoilt in the south. Theatres outside of London can do one show in their season that has a cast over 10, and that's not a great breadth of work to be doing."

If Morrissey was wealthy before – he lives in, by all accounts, a very nice house in Hampstead – landing a big US TV show will have catapulted him into a new league. Can money make him feel disconnected from his roots, his siblings? "The great thing about coming from where I come from – Liverpool and my family – is that we're very close. I have a great relationship with my siblings and their kids. I don't feel I live in a rarefied world in any way. When I go home, people are very vocal in telling me what they think. I don't feel disconnected in that way, and also my work tends not to make me rarefied. If you're doing something like Red Riding, the places we film in, you're investigating the real world there. I grew up in a great, loving place, and I try to recreate that; the difference, for my kids, is space and stuff like that."

He says journalists often ask him about his marriage into the Freud dynasty "with the type of tone of 'that snotty-nosed oik from Liverpool who ended up … '", but anyone who has read Hideous Kinky, Freud's autobiographical first novel, based on her bohemian and at times impoverished childhood, may make you doubt money has ever been much of a point of difference between them. But I wonder if he spends any time marvelling at the idea of having children who are also descended from two 20th-century giants (Sigmund is their great-great-grandfather, and Lucian, who died last year, was their grandfather) – and whether he thinks it could be a burden. "I don't think they have a sense of that, really," he says. "They had a relationship with Lucian, which was great, but I don't think they see themselves in terms of their ancestry. If they do, they don't talk to me about it."

Americans, he says, are more bowled over by Sigmund; to Morrissey, Lucian was more interesting. "I wasn't very close to him, but whenever I met him, I found him endlessly fascinating. I've been in his studio when he was painting. He used to do this thing where, as he painted, he would just put it [surplus paint] on the wall – the wall was that thick with paint." He holds his hands a foot apart. "I would look at that and think it was amazing."

Morrissey is a collector of such details. When I ask him what he likes about acting, he says, "I like the fact it gives me the opportunity to examine other lives." For someone who claims not to have been academic at school, he takes a rigorously academic approach to his work – roles are researched thoroughly, through books and interviews, until he has absorbed not only the personality he thinks they have, but the time and place they were living in.

He has directed a feature film – 2009's Don't Worry About Me, set over a day in Liverpool, as well as some short films – and would like to do more, but acting is his first love. "I love telling stories. I like the challenges presented to me on a daily basis. There's nothing resting about acting. There's something I must love about the insecurity I profess to hate." Insecurity about whether the phone is ever going to ring again? "Yes, but mainly about self, of doing something and going: 'Was that OK?' All actors have a great level of insecurity, which can be really boring, particularly if you're on the outside of that – it can seem egotistical. Sometimes you're going: 'Why didn't I do that other job? Why is he doing that job when I'm not?', and if you're not careful it can drive you mad. My most creative time is in the car from the set at the end of the day: 'I could have done that! Why didn't I do that?'"

Still, he says with a slow guilty smile, he thinks he must secretly enjoy the self-flagellation. "I do give myself a bad time, but I sort of like it. I'm not a perfectionist at all. I find perfectionists boring because the real creative heart is in the mess somewhere."

The Walking Dead is on FX, Fridays at 10pm