On a fiercely sunny afternoon in London, Domhnall Gleeson lopes into the hotel courtyard in a blue hoodie and grey jeans, and sticks out a hand to shake mine. The cheerful 29-year-old pulls up a chair in the shade; he is pink and tall and skinny, with a shock of short orange hair in place of the lank locks worn in some of his most widely seen work. (He was Ron Weasley's scarred brother Bill in the final Harry Potter instalments, and a whimpering outlaw in the Coen brothers' True Grit.) He is well known for his well-known father, Brendan Gleeson, who makes such a feast of supporting parts that you wish he got more leads. The same can be said of Domhnall (ignore the 'm' and rhyme it with 'tonal').

Though not, perhaps, for much longer. The haircut is for a new movie – Richard Curtis's latest rom-com, About Time, in which Gleeson plays the romantic lead, a time-travelling lawyer, opposite Rachel McAdams. Curtis films tend to do miraculous things for an actor's profile: think of Hugh Grant, Rhys Ifans, Bill Nighy. Does Gleeson feel any trepidation about this? "Yeah, yeah," he says decisively. "I'll just grow my hair long when the film opens. I'll cope. I mean, I'm not built like a leading man. Richard's taking a bit of a risk casting this face in that role. Conventionally handsome is not really where I'm at."

The Irish writer-director Tom Hall, who cast Gleeson in Sensation as a rural lad turned ponytailed pimp, agrees with the actor's assessment of himself.

"Domhnall has a very mutable quality," he explains. "He can look really beautiful on screen one minute and really alarming the next. It makes him very versatile. He gets offered lots of cowards, creeps and sex pests, but he's also an appealing and unconventional romantic lead."

Gleeson will shortly be seen in two new and dissimilar adaptations: the science-fiction thriller Dredd ("It's kinda great and pretty full-on," he enthuses), based on the 2000AD comic strip, and Joe Wright's lush adaptation of Anna Karenina. In the former, he is a computer whizz whose eyes have been replaced by cameras; in the latter he plays the impassioned nobleman (and Tolstoy surrogate) Levin from behind a copper-wire beard. This month, he also appears in Shadow Dancer as Connor, an IRA soldier whose sister (Andrea Riseborough) turns informant. With his furtive eyes and ghostly pallor, Connor has been shaped and calcified by vengeance, and Gleeson uses the few scenes he has to make him achingly real.

"Portraying as human the people you hear about on the news doing bad things is dangerous," he reflects. "But it's also necessary and important. And even more important is that the movie is entertaining, which I believe it is. Anything else it's trying to do is bollocks if it's just a statement. I don't want to watch a statement."

With the exception of Sensation, and the Live Aid drama When Harvey Met Bob, in which he plays Bob Geldof, Gleeson's career has been restricted to detailed miniatures. "I'd rather do one day on a really cool movie than six months on something crap," he explains. "I'm not going to get better as an actor working on bad stuff." When he wants a particular job, he says he goes "hell for leather". One audition tape shot on camcorder in a cottage in Ireland won him the part in True Grit; another attracted compliments from Todd Solondz, who is high on his wish-list of directors to work with. So too is Terrence Malick. "Rachel McAdams just did a film with him so I've been annoying her on set by going, 'Now what would Terrence do in this situation?' and pestering her for stories."

Gleeson's short, spotless CV looks especially impressive for someone who doubted he wanted to be an actor at all. His father ditched a teaching job at the age of 34 to pursue acting but Gleeson, the oldest of four boys, has no recollection of tough times in his Dublin childhood. "You might know you didn't have the coolest trainers but you don't realise it was hard for your parents to get Christmas together some years." As a teenager, he gravitated toward writing and directing, determined he didn't want to be defined by his parents: "No one does." He refers to his father admiringly and often, but he'll bristle if someone tries to use that relationship as a stepping stone. "I get asked to give stuff to my dad," he splutters. "I'm, like, 'I'm not gonna pass your script to him!' You know? My dad's my dad. I'm not his agent.'"

At 16, Gleeson collected an award on behalf of his father; he delivered an acceptance speech full of chutzpah and promptly found himself with an agent. By 19, he was raising the roof in the West End in Martin McDonagh's raucous IRA-themed comedy The Lieutenant of Inishmore. Four years later, he went with the play to Broadway, where he was nominated for a Tony award. But the memories of agonising stretches of unemployment stayed with him. "As an actor, there's very little you can do if people don't want to see you. Just getting yourself into the room to audition is tough."

His father had told him that when you're not getting work, you have to make your own. The upshot of this is that Gleeson is now a filmmaker himself, with two shorts, both festival hits, to his name. What Will Survive of Us is a clammy comedy about romantic idealism and anal sex. "I thought it'd be funny," he laughs. "But it's a bit embarrassing when your Dad puts it on in front of your uncles on Christmas Day." For the follow-up, Noreen, he cast his brother Brian, also an actor, and his father as dimwit policemen who turn an open-and-shut suicide into a tableau of Jacobean carnage. Both shorts are characterised by a wistful comic despair; they bode well for the full-length feature he has been writing for the past two years.

It might have been finished by now if the acting career hadn't taken off.

Following a few months at the start of 2011 that brought the release of True Grit, Sensation and the film of Kazuo Ishiguro's novel Never Let Me Go, in which Gleeson played one of a community of clones, he could scarcely leave his home without having an award thrust into his hands. He was named a Shooting Star at the Berlin Film Festival, and a Rising Star by the Irish Film & Television Academy. (Collecting his son's award, Brendan Gleeson joked: "I'm not so sure about 'rising star'. I could never get him up in the morning.") IFTA also made him Best Actor for When Harvey Met Bob, though Peaches Geldof provided a lone voice of dissent. "She hated it," laughs Gleeson. "Which is completely cool. She said I had a leprechaun's accent. I'm bloody Irish! But no, I respect that. I mean, he's her Dad."

Those who know Gleeson are confident in his future. "I've kept in touch with Domhnall since Never Let Me Go," says Kazuo Ishiguro, "and I've become really struck not just by his huge and subtle talent, but by his modesty, and the decent way he instinctively treats everyone around him. It's unusual to find a young actor on the brink of stardom so useless at self-absorption. Instead, I get the impression he's always watching and thinking about everything going on around him, with a real understanding of people, and this should serve him well not just as an actor but with his directing and writing ambitions."

"Domhnall is probably in two minds about the higher profile he's going to have soon," says Hall. "Lots of young actors in that situation could easily have their heads turned, but his instincts have been unerring." For now, Gleeson is having fun with the anonymity he enjoys. (Except in Ireland, where his larky work on the profane TV sketch show Your Bad Self has made him the sort of star who gets his own punchlines quoted back at him by students.) He tells me he has lost count of the number of strangers who have approached him on location during About Time to ask whether there is anyone famous in the film. "I usually say, 'Nah. They've got some crap guy off the telly, but apart from that …'"

• Shadow Dancer opens 24 August, Dredd and Anna Karenina on 7 September. About Time is released next March.

• This article was amended on 17 August 2012 to correct the spelling of Brendan Gleeson in the standfirst.