The biggest band you've never heard of? Try Paris-based Lilly Wood and the Prick. This summer, the remix by German electronic music DJ Robin Schulz of their single, Prayer In C, reached number one in 35 countries, including the UK. Meanwhile, the video on YouTube has garnered upwards of 48 million views. You would imagine Lilly Wood's singer, Israeli-born Nili Hadida, might be ecstatic. Not quite. "I have a very hard time being alive," says the 28-year-old on the phone from her home in the cosmopolitan 20th arrondissement of Paris. "I'm not a very happy person. I'm not very good at life." It runs in the family, apparently.

"My dad is the most depressed person I know," she continues. "I feel terrible because every time I speak to him there's nothing I can do." What is he depressed about? "He's depressed because he didn't bring me up, because he's not doing a good job with his family, because he didn't manage to become very rich - everything!"

If there is a streak of melancholy running through dance track Prayer In C, it doubtless comes from Hadida. She wrote the lyrics, after all. "See our world is slowly dying… Yeah, our hands will get more wrinkled/And our hair will be grey/Don't think I could forgive you/And see the children are starving/And their houses were destroyed/Don't think they could forgive you," she sings in her lightly raspy voice.

As good-time club-based party anthems go, it's pretty dour, full of apocalyptic imagery and portents of doom. And asked what informs the songs on Lilly and the Prick's albums - 2010's Invincible Friends and 2012's The Fight - she replies, "Pain in general". Rising antisemitism in France, suffering in Africa, where she spent a month last year recording her forthcoming third album, it all feeds into the music she makes with her partner, guitarist Benjamin Cotto. "I think life is horrible for everyone," she decides. "I feel bad about all of it."

Unexpectedly, Lilly Wood and the Prick don't sound like Leonard Cohen on downers. Despite the titles - Hopeless Kids, Down The Drain - these are perky, catchy indie pop ditties that just happen to grapple with the dark side of the human condition. You could even call them commercial.

My dad is the most depressed person I know

"Well, 'commercial' is a big word," she reflects. "But, yes, the music is easy to listen to and like." Hardly surprisingly, the "up" quotient is courtesy of Cotto. "Ben symbolises the happy side of our music," she confirms. "It's funny because our songs always sound happy, then you listen to the lyrics and it's like, 'Oh my God, this girl is really depressed'."

Hadida was born in Tel Aviv in 1986. After splitting with her father, her mother moved to Paris when Hadida was three months old.

Aged 14, she spent a year-and-a-half at a boarding school in Sussex, after which she lived for two years in California with her father, a pioneer in web programming and something of a star in the world of martial arts. He now lives in Israel with his second wife, their son and two daughters - one an Israeli Army officer, the other about to start her national service.

"I was in Israel last year and people into martial arts were queuing up to have their picture taken with him," she marvels. Her maudlin spirit can perhaps be attributed to her family background - both her grandfathers lost their lives in combat.

"My mum was born in Algeria right before they achieved independence," she explains. "Her dad was murdered there. He worked with Arabs so people hated him. Someone put a bullet in his head. My dad's dad died in a very famous battle in Algeria. And I didn't have my father growing up because my parents separated, so I'm a typical daddy-issue girl who's about to turn 30. So unoriginal, right?"

Her mother, born Jewish, is now a Scientologist. Hadida describes herself as atheist. "I don't believe in anything," she insists, adding: "I love my origins, but I'm not a religious person. I'm Jewish and sometimes I do Yom Kippur, but I'm more a person of the world than anything else." She didn't have a batmitzvah, although her two half-sisters in Israel did. She feels bad for the youngest one, about to enter the military, even though she appreciates the necessity of army service.

Hadida is down on most things, including her countrymen ("French Jews are so obnoxious - they give us a bad name").

But she rates Tel Aviv "the most open-minded city in the world" and its inhabitants "the smartest people I've met in my life".

Having "hated being a kid", she says she "wouldn't want to bring a child into this world". Yet she is stepmum to her boyfriend's five-year-old son. She might profess to being miserable but it gives her a certain energy. She needs it, what with Prayer In C having been a hit in so many countries - including "really random" places, she laughs, such as Zimbabwe - and the demands that success brings. For example, she and Cotto have been invited to appear on Italy's version of The Voice.

How has success changed her? "So far, not so much," she replies. "I guess my bank account is going to be a lot fuller next year." It's a shame, I suggest, that they haven't found a way to monetise all those YouTube views. "Oh, we're not in it for the money," she replies. "We just want recognition for the music."

That might involve changing the band's name. The Lilly Wood part - apparently from a dream she had about Ron Wood in which the Rolling Stone had a daughter called Lilly (go figure) - is fine.

But the Prick bit, which they came up with in 2006 when they formed the band, will have to go.

"What can I say? We were 20 and found it funny," Hadida says by way of justification. "We don't like it too much, either. We're bad at choosing names in general. Our album names are s***. Invincible Friends? That's so cheesy."

Maybe the best thing you can say about Hadida is that she's unrelentingly realistic.

She accepts her and Cotto's limitations and mistakes and embraces the "cheesiness" of the '80s pop that influences their infectious melodies. For her, honesty is the only policy. "We're not trying to be cool," she declares. And for once she sounds happy.