From hipster crush on the US comedy scene to making it big with her first major film role: deadpan TV star Aubrey Plaza talks frankly about having a stroke, working with Charlie Sheen and why she's nothing like her character in Parks And Recreation

When Aubrey Plaza opens the door of her new house in the foothills of Los Angeles' trendy Los Feliz neighbourhood one rainy late afternoon in November, she is dressed for a cosy night in: leggings, over-sized sweatshirt, slippers.

With her Betty Boop pout and deadpan delivery, the Delaware-born daughter of Puerto Rican and Irish parents has been a hipster crush since she exploded on to the US comedy scene in 2009 with roles as a stand-up comedian in Judd Apatow's Funny People and as a surly city government employee in the NBC mockumentary sitcom Parks And Recreation (coming to BBC4 next year). But Plaza has never played a mere girlfriend or stock ingénue role, and admits she's less than comfortable with the idea of selling herself on her looks – for example, by posing for the photos accompanying this article.

"It's weird, because that's a job that some people have – they're called models," Plaza says as she curls up in a wooden chair at a kitchen table dominated by a deluxe Scrabble board while her big, friendly dog, Stevie, nuzzles her legs. "It's a skill that not everybody has. There's no photo-shoot academy. If there was, I'd probably be kicked out." As she gets up to put on the kettle for tea, she acknowledges that posing prettily is an unavoidable part of her chosen job. "I try to make it fun," she says. How? "Drugs and alcohol," she says without missing a beat.

In Parks And Recreation, Plaza's character, April, is described by curmudgeonly boss Ron Swanson as "both aggressively mean and apathetic". He pauses and beams. "She really is the whole package." The character's admiration reflects the real-world response to Plaza: she's a certain kind of guy's dream girl, authentically world-weary in an industry whose idea of a young female misfit is Zooey Deschanel.

Plaza's character in Funny People was partially based on Janeane Garofalo (who worked with writer/director Apatow in the early 1990s on The Ben Stiller Show), and the younger actor/comedian has continued to draw comparisons to the Reality Bites star throughout her brief career. But Plaza represents several evolutions down the line of women who use their smart mouths to critique an industry that treats pretty girls as a usable resource. Part of her appeal is the way she wraps up her star power in a blanket of knowingness so dry that it almost makes you forget about the way she looks.

Her parents were 20 when she was born – "I was, like, a big accident," she says. "We started out with nothing, but my parents worked really, really hard, and we kind of moved up. My mom is a lawyer now, and my dad's a financial advisor." She is the eldest of three sisters.

At a young age, her mum introduced her to Saturday Night Live. "I became really obsessed with comedy and improv," she says. "I wanted to be on that show for as long as I could remember." Plaza was not an idle dreamer: as soon as she identified SNL as her perfect job, she set to work investigating how to land it. "Just by looking at people's biographies online, I saw that a lot of them did sketch comedy. And I decided that's what I was going to do." In high school, she enrolled at an improv school in Philadelphia, the closest major city to Wilmington.

After school, she moved to New York to study film at NYU and take improv classes at the Upright Citizens Brigade. Soon she was performing with the likes of Rob Riggle (The Daily Show), John Mulaney (later SNL's head writer), and her current Parks co-star, Aziz Ansari. She interned at Saturday Night Live in 2005, and then co-starred in a web series, The Jeannie Tate Show, produced by Maggie Carey, wife of SNL star Bill Hader. In 2008, she finally landed an audition for SNL – and didn't make the cut.

"Then I was offered the part in Funny People." Plaza moved from New York to Los Angeles to shoot the Apatow film, assuming that she'd return to New York when the job ended and resume her place on the East Coast comedy scene. "And I never left here. I was cast on Parks And Rec and Scott Pilgrim Vs The World, and it was all happening back to back. So I was thinking, 'After this, I'll go back.' And then Parks just kept going. Which is a good thing, but it was kind of abrupt."

This sudden rush of success is all the more remarkable considering that in July 2005, when Plaza was 21, she suffered a stroke. She returned home to Delaware, where she underwent speech therapy, and battled anxiety attacks. At the end of the summer, her parents tried to convince her to take some time off, to focus on her health, but Plaza was determined to get on with her life. "It was one of the scariest things that's happened, but it never felt as if it was keeping me from doing anything. "If anything, it made me more aggressive about getting things that I want. Maybe subconsciously I was feeling it was… not a near-death experience, but it put things in perspective."

Plaza wouldn't have been able to accomplish so much so quickly if not for unusual tenacity and ambition. Her agent, she says, saw her in The Jeannie Tate Show, "agreed to meet me for coffee, and I just harassed her. I sent her everything and eventually she started sending me on auditions.

"I think people just associate me with my characters," Plaza says. "But in high school I wasn't like that at all. I was president of everything I could be president of. I wasn't, like, the weird one making comments at the back of the class. I was at the front of the class. I was annoying."

On the road: Plaza in new film Safety Not Guaranteed

Her new film, Safety Not Guaranteed, features the 28-year-old in her first starring film role, and both plays on the expectations Plaza's fans may have of her and challenges them. She plays Darius, a graduate intern at a Seattle magazine who is still recovering from the death of her mum 10 years earlier. Desperate to, as she puts it, "eject" from her rut, Darius jumps at the chance to accompany two co-workers on a road trip to investigate a classified ad posted by a mystery man who claims to be in search of a partner for time travel.

The would-be time traveller is Kenneth, an intense 30-something grocer's clerk who claims to have built a time machine in his garage. Darius's boss convinces his intern to use her feminine wiles to wedge her way into Kenneth's life. "You're dangling my vagina out there like bait," Darius complains, but she gets in so deep that she falls in love with her mark.

The film was inspired by an ad in a publication geared to rural American survivalists, which then became an internet meme. Plaza was attracted to the idea of taking such a potentially easy premise for comedy and bringing depth to it: "It's such a thing now, people making fun of other people on the internet." It's a thing she has been a part of – Plaza's impression of "controversial" comic Sarah Silverman has racked up nearly 750,000 views on YouTube since she uploaded it in 2008 with the disclaimer, "I LIKE SARAH. I am a fan of hers. For all you haters." Today, she says, "It's a harmless thing, but I think the movie is challenging the idea of that, and the idea of people being so ironic about everything all the time."

The chatty comedy builds to a spectacular sci-fi climax that, despite its charmingly lo-fi production values, delivers a transcendent emotional wallop. The director, Colin Trevorrow and his star bonded over their shared love of fantasy classics such as Back To The Future and Jurassic Park. "A lot of independent films are harsh reality checks," Plaza says, "but this is an independent film with a blockbuster feeling."

Trevorrow says he and Safety's writer, Derek Connolly, "were told we could get millions more to make the movie if we recast it with bigger names. Derek and I both agreed it had to be Aubrey, and never backed down." Part of the impetus for casting her came from wanting to perform a kind of bait-and-switch on audiences expecting to get a rendition of one of her cynical characters. "A lot of people think Aubrey has one note that she can play, and we used that to our advantage."

As Plaza puts it, "Sometimes I'm viewed as the poster child for irony, so it was cool to play against that. My job was to go on that journey, and allow someone who is pure and sincere to open me up to things." That same process of transformation, she says, "is what we want the audience to feel, in the end. That it's OK to believe in positivity." She cracks up a bit on that last word, as if she is unable to argue against irony with a completely straight face.

But she seems sincere when she says, "I love great acting, as nerdy as that sounds." Her idols and inspirations tend toward the old school. She gushes about Judy Garland: "I love her as an actress. She's amazing, and her voice, obviously, it's like no other voice ever. And she's funny!"

Does Plaza ever feel that she doesn't get credit for her own work, that people just assume she's playing herself? "Yeah!" She giggles. "But also, I don't blame them. In some ways, that's the ultimate compliment. If people think that's me, I'm really doing it well. But the movies are opportunities to show off other things I can do."

She tries to spend all of her time off from Parks on movie sets. She'll soon be seen in Roman Coppola's A Glimpse Inside The Mind Of Charles Swan III, co-starring opposite Charlie Sheen, which she calls "a crazy experience".

"I played his assistant, so it was mostly me yelling at him to get his shit together. So that was fun," she laughs. "I learned a lot from him. He's so natural, and so professional. Half the time I wouldn't know if he was talking to me or we were doing the scene… He's fun to be around. He's always telling crazy stories. And he doesn't care what anybody thinks about him. He's the ultimate cool guy."

She's also preparing to star with John C Reilly in Life After Beth, a zombie comedy written and to be directed by her live-in boyfriend, Jeff Baena (co-writer of I Heart Huckabees). She sums it up with a wry smile: "It's a really fucked-up movie."

Plaza is quick to admit that she doesn't get every part she wants. "People don't totally trust me yet," she laments. Safety Not Guaranteed was the rare film she was offered outright. "I still have to go after things that I want," she says, and sometimes indefatigability isn't enough – she had to pass on one of the leads in Whit Stillman's Damsels In Distress when she couldn't fit the movie shoot into her TV schedule.

"I want to be considered a well-rounded actor, not a weirdo," she says. That said, "You can't control what people think. And I can't not have some parts of what I'm doing be me, because it's all I have to use. I'm not, like, Daniel Day Lewis." She grins. "Yet. I will get there!" Despite the self-protective tinge of ironic self-awareness in her voice, I have no doubt she means it.

• Safety Not Guaranteed is released in cinemas nationwide on 26 December.