It is a typically sunny Los Angeles day and Karen Gillan is nursing her morning coffee, feeling mildly smug.

"I'm going to incorporate sunshine into every answer," she informs me, well aware that anyone reading this back home in Scotland may not quite be experiencing the same temperate weather conditions. She can't get too cocky, though: her typically Celtic blend of pale skin and red hair is not designed for sun-worshipping. "I'm ginger!" she cries. "I can't be in the sun."

All of which means fans of the 26-year-old former Doctor Who star won't find her on Venice Beach. "I don't sunbathe," she says, unsurprisingly. Not that it matters, for a sizzling career means she's more than having her moment in the sun. She's been living here for the past few months, since completing her new film Oculus, a creepy low-budget horror that's her first venture into Hollywood. "I thought, 'I might just stay for a while and see what happens,'" she says.

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What's happened is a gradual ripple effect from playing Amy Pond, the perky assistant to Matt Smith's Tardis-flying Time Lord in Doctor Who. Take the director of Oculus, Mike Flanagan. Gillan first met him on Skype. She was in her "childhood bedroom" at her parents' house in Inverness, shortly after she moved back to Scotland in 2012 after selling her house in London. Adorned with Muse posters and a Daniel O'Donnell calendar, it could've been an embarrassing locale for a first chat, were it not for the fact Flanagan is a massive Doctor Who fan.

As soon as they hooked up online, he was swigging from a Tardis mug. "He didn't realise and I didn't say anything. But I thought, 'I'm probably going to get this role.'" She did, and they met again in LA. "He came to talk to me and he was wearing a T-shirt with Matt Smith's face on it. And then, to his horror, Matt Smith joined us. I was like, 'Do you mind if a couple of my friends come along as well?' And it was Matt Smith and Arthur Darvill [who played Rory Williams in the show], and he was like: 'I can't believe I'm wearing this T-shirt.'"

Although briefly appearing as a soothsayer in an episode a year earlier, Gillan joined the show as Amy in 2009, making an instant impact when she arrived on screen as a kiss-a-gram dressed in a policewoman's uniform. The tabloids had a field day, renaming the show Doctor Phew, while Smith dubbed her the "sexiest companion ever", putting her ahead of Billie Piper and others. She stayed on the show for three years - generating legions of devotees.

Gillan followed it with a stage bow at London's Donmar Warehouse in John Osborne's play Inadmissible Evidence and a turn in last year's breezy Glasgow-set romcom Not Another Happy Ending. Making the move to the US for Oculus, though, is a major step forward. "I love horror films so much … I like being scared," she says. "I don't know what that's all about." She giggles; something she does a lot. "I want to be in good films, obviously, rather than cheesy ones. And some horror films can be quite cheesy. I love watching them but I don't necessarily want to be in them."

In other words, you won't find Gillan running around in a skimpy wet T-shirt. "I don't need that in my life," she nods. Playing the opposite of your typical scream queen, Gillan's character Kaylie Russell confronts Oculus' supernatural phenomenon head-on: an ornate mirror with the power to make those in its vicinity do all sorts of nasty things. Her mission is to destroy the looking-glass; one she believes cast a spell over her brother Tim, who was committed to a psychiatric unit 11 years earlier after a murderous act.

Within a few hours of Tim being released, she's enticing him to return to their family home and help destroy the mirror. "That was what was most appealing to me about the character," says Gillan. "We are accustomed to having a female lead in a horror film; that's quite typical. She's always running away, whereas this character is running directly to the threat. And the worse things get, the happier she gets. She doesn't look at the mirror as a symbol of fear or threat. She looks at it as a symbol of hope, because that's what she can use to prove [that her brother is not insane]."

Apart from a suitcase full of clothes and a working visa, Gillan travelled light when she went to Alabama to shoot Oculus. "I didn't bring any of my possessions," she says. "They're all still in London, in a storage unit, and there's loads in Scotland too. I had to start from scratch." There must be something freeing about that; a reinvention of the self. Rather like her current hair style. As just about anyone who has seen a picture of Gillan in the past 12 months will know, her distinctive flame-coloured locks have been shorn.

It's all in aid of her second film of the summer, Guardians Of The Galaxy, in which she plays a bald, blue-skinned space pirate named Nebula. Last year, footage of Gillan at the 2013 Comic-Con event in San Diego saw her arrive on stage in a pair of denim shorts, daisy print T-shirt and - fooling everyone - a red-haired wig. "I might have done something a little bit crazy," she announced on the panel, whipping off the hair-piece and throwing it in the audience to roars of approval.

Coming from the ultra-successful Marvel Comics stable, it's from the same universe as The Avengers superheroes - Benicio Del Toro's character The Collector was glimpsed in the post-credits sting of the recent Thor: The Dark World - but will tell a story that's largely independent of the world inhabited by Iron Man, The Hulk et al. Telling of a gang of misfit alien space rangers, it's as if the cast-offs from the Star Wars cantina got together for a jaunt in a galaxy far, far away.

"It's a real change in direction for Marvel, in my opinion," says Gillan. "Just the tone of it - it's really, really funny, [and has a] cool soundtrack. I didn't know much about the franchise. So I went back and read some of the comics. It's just amazing. There's a racoon and a tree, in leading roles." That would be Rocket Racoon (voiced by Bradley Cooper) and Groot (Vin Diesel) - two computer-animated characters that makes up two-fifths of the Guardians.

When it came to chopping off her locks, it was liberating, she admits. "Everyone has a feature that the people use to describe them - the girl with the red hair, or the really tall guy. So then to get rid of that feature and not be that person any more is an interesting concept." Was it traumatic? She shakes her head. "I just sat in a chair and they started shaving it off my head, and then I fell asleep for two hours. But when I woke up, I was a bit like: 'Oh my God, where is all my hair?'"

I mention that those early stills of her with an entirely shaven skull made her look like that other famous sci-fi heroine, Sigourney Weaver's Ripley, in Alien 3. "I will take that," she beams. Now, a year on, it's gradually grown back in - leaving her with a pixie Audrey Hepburn-like crop that presents its own set of problems. Giving her the look of "a little boy", as she puts it, it's not quite like her long flowing locks of old. "I don't know what to do with it. It sticks up when you wake up."

While one recent pap shot saw her leaving an Agent Provocateur shop with "bedhead hair", as the caption put it, it's a small price to pay for landing the role of one of the film's major villains, banishing Gillan's sweet-natured girl-next-door image for good. "It was a really cool transition," she says. "My Dad has always said: 'You don't want to play a Bond girl. You want to play a villain.' Basically, don't just settle for running around in dresses. I was like: 'Woo - feminism, Dad.'"

He may not be in the business, but her father, John, was an amateur singer, performing at open-mic nights in local pubs. "We love a bit of karaoke," she explains. "We call it Karen-oke." Her own number is Fever by Peggy Lee "in a full Scottish accent". That's pretty sultry, I say. "Not with the Scottish accent," she deadpans. While she was initially shy when it came to performing, music came early in her life, after her father introduced her to the work of Ella Fitzgerald, Nina Simone, Elvis Presley and Frank Sinatra. "That was my first love," she says.

Gillan, an only child, was born in 1987 and was always smart for her age. Her father taught her to read and play chess before she went to school, and she learned to play the piano by the age of seven. If being a singer was her first ambition, she switched her attentions to acting after spending time in local youth theatres and drama clubs. "I was like: 'I'm going to run with this instead.'"

At school, she was "that weird, long, ginger girl"; but does that mean she was picked on a lot? "I wasn't horrifically bullied. There was name-calling but nothing awful. All kids get teased about something. And this was quite an obvious feature. When you're really tall and ginger and white at school, you're going to get it." Still, it might account for why she dyed her hair pink when she was 16, even if, on the surface, it was in honour of a Beyonce video.

Along with a belly-button piercing, that was the extent of her rebellious teenage angst phrase. She was too driven, heading off to Edinburgh to complete an acting and performance course at Telford College. A year later she was in London at the Italia Conti Academy, but a few months into her intended three years, she dropped out to take eight days filming experience on one episode of Scottish detective show Rebus.

"I was really restless, because I wanted to be acting," she reasons. "But I was thinking: 'Was it the right thing to leave drama school for this one episode?' Whereas I could've stayed there for three years and got a degree. But I don't know. I'm like that. I just left randomly. I was longing to get back into acting, but it was quite difficult." To make ends meet, she started working in a pub, The Pilgrim in Kennington, South London, until she was scouted by a modelling agency.

It never sat well with her. "I always resented the fact I did modelling. I associated it with not doing acting. If I was doing modelling that meant I wasn't acting." Still, standing at a slender 5ft 11in - only just smaller than Matt Smith, who cheekily likened her lithe limbs to a preying mantis - she comes ably equipped to do the job. "I never aspired to do modelling," she reiterates. "It was the equivalent of me working in a pub on the side." But better paid? "That's why I did it."

One of her first jobs as a model was participating in the launch party for Dainty Doll, a make-up range endorsed by that other famous redhead, Girls Aloud star Nicola Roberts. In the show, aptly enough, she played a red-carpet-strutting character called Premiere Girl. But she was always hustling for acting jobs. "My agency hated me because I was always like running away from things to try to get auditions. I was really bad."

Soon enough, she was gaining bit-parts on TV dramas, before landing a two-year stint as part of the sketch comedy series The Kevin Bishop Show. Impersonating the likes of Angelina Jolie and Katy Perry, it wasn't exactly Monty Python, but it gave Gillan some much-needed experience. "I got to dress up and do loads of different characters and accents, and be pretty ridiculous and get paid for it." Far better than modelling then.

When she was cast in Doctor Who she "totally quit modelling for ever" - although it didn't stop her picture being taken; the paparazzi descended on her, even following her to a friend's wedding in the Highlands. "There was a point in my life where it was happening every day. They would be outside my house every day - I was living in this horrible little flat - and they wouldn't even leave their cars. They would just follow you around, taking pictures from inside the cars. So you wouldn't even know they were there."

She calls it an "extreme" time. "I was quite young and I wanted to go to the pub with my friends, and not worry about having one too many." And there were plenty of snaps of her looking worse-for-wear after a night out. At least she had photographer Patrick Green by her side. "I had a long-term boyfriend for that entire period, so that was quite good and stable," she says. "He would get followed a little bit at the beginning. He was fine with it."

She and Green, who were together for six years, split in 2012; since then, she reportedly dated Stanley Weber, the handsome young French actor from Not Another Happy Ending, but has stayed quiet on the romance front. Thankfully, it's calmed down for her since those early days on Doctor Who (she didn't even appear in the 50th anniversary episode of the show last November). "It's not anywhere near as crazy as it was at the beginning. I think people [initially] are like 'Oh, who's this new person?'"

With her move to America, when it comes to preserving her anonymity, it'll help that her upcoming career-making role will see her in disguise. "I doubt people will be like: 'Aren't you the blue, bald woman from Guardians?'" she chuckles. However, already slated to star in Selfies, an upcoming US show for ABC, that may change. She'll play a social-media obsessed gal named Eliza Dooley, a modern-day spin on My Fair Lady's Eliza Doolittle.

She'll also return to horror in Ti West's In A Valley Of Violence, co-starring with Ethan Hawke and John Travolta, and then there's The List, a rom-com in which she plays a woman who gives her boyfriend a run-down of all the things she'd like to change about him. "They wanted me to be Scottish, which is ace," says Gillan, who claims she would never follow her character's rather extreme demands. "I feel like if you have to give a man a list, maybe he's not the right man."

Her coffee drained, our time is up, with Gillan readying herself for a busy day ahead in the LA sunshine. She has a lot to do and needs to get her head together. "I'm the least organised person in the world," she trills. Full of the joys of early summer, the happy-go-lucky aura she gives off suggests that she doesn't plan too far ahead. As long as she remembers sunblock, she'll be fine. n

Oculus (15) opens on June 13. Guardians Of The Galaxy (TBC) is released on July 31.