The bunny-boiler's book: Jessie Cave interview

Endlessly perusing others' social media profiles might now be so ubiquitous it's thought normal, though few would want their online voyeurism made public. Moreover, there can't be many who'd make a drawing of this kind of behaviour and post it on Twitter daily, but back in 2010, Jessie Cave began such an activity: "It's like a diary," she says, "I can look back at the pictures and know what I did that day."

Love Sick is the book of Cave's drawings. The sketches within its pages are deliberately child-like, with Cave's characters in the midst of seemingly-innocent everyday exchanges. They might be at the bus-stop, a cafe or meeting someone for the first time: 'Hey! I've already stalked you extensively on Instagram. Nice to meet you,' says one meeting another for the first time. Rendered in a nursery colour palette, the crayon style drawings wouldn't look out of place on a proud parent's fridge door. In reality though, these exchanges explore envy, anxiety and insecurity at all the stages of a relationship – in other words, they're about lovesickness.

Though it may be the love of aching, longing and confusion, it is still love Cave has on her mind. Especially in the way the drawings can relate to people: "The reason I do these drawings is so people know we're all in the same pool. We all have anxieties."

Cave is, of course, still best known for her role in the final Harry Potter films. In Harry Potter and the Half-Blood Prince, she memorably owned the role of Lavender Brown, giving a ditzy character archetype something of a Fatal Attraction edge as she pursued Ron Weasley. Despite the fact that she and Lavender seem to share similar obsessive characteristics, Cave finds the idea she developed anything of Lavender's personality into her own onstage persona somewhat amusing. In fact, cause and effect are the wrong way round. There was something already about Cave that suggested she was perfect for Lavender: "I was never meant to get the role," she says, remembering that at the time of her audition she was 20, perhaps too old – relatively speaking – to play a 16 year old. However, on the day Cave had to do a little improv: "I made Rupert [Grint] laugh. I'm pretty sure that's why I got the part." It was then Cave's comedic abilities then that went into Lavender, rather than Lavender's character influencing Cave's act.

"I've never pitched myself as the Harry Potter actor," she says. Although she recalls in Bookworm, her 2012 debut Edinburgh show, "I did pretend I'd broken into Leavesden studios to get the role." Refreshingly, Cave never bores us with some monologue about wanting to be taken seriously or distance herself from Lavender: "It was the best thing to happen to me," she says, and is cheerfully aware that, "even if I'm a politician people will always ask and comment, 'where's Won-Won?'"

I Loved Her: Jessie Cave at the Fringe

Her new Fringe show I Loved Her sees Cave back at Underbelly for the first time since her debut. Bookworm was a popular show and, if comedy reviews are anything to go by, a well received one. Yet twice in our conversation Cave sounds like she has unfinished business, which she hopes her new show will take care of. She's hopeful I Loved Her will be something of a "mature version of the first show"; the end result of what she's been working towards.

Not that Cave hasn't returned to Edinburgh every year since 2012. In the following year she shared an hour with musical comedian Jenny Bede. Then last year there was the sketch show Grawlix with talented TV actor and writer Emer Kenny – or at least, there was meant to be.

Unfortunately Kenny had to pull out of the Fringe at the last minute, leaving Cave to her own devices. Cave chose to keep the lunchtime slot at Espionage and try out some solo material. She decked out her corner of the bar like a children's play area and, in a characteristic move, made it easy for the unwary to believe this would be a fluffy Fringe hour to turn the brain to candy floss. Of course, the resulting show was no playschool. Cave's creepy material – enveloped in her softly spoken delivery – settled uncomfortably under the skin.

One notable aspect about last year's show, and one which Cave continues to develop, is her use of rudimentary puppets for dialogue purposes, with faces drawn on paper plates and held up at her eye level with a stick.

"I'm most comfortable with dialogue," she says, "and with puppets it's a way for me not to be alone on stage." Cave also promises to enhance this aspect of her performance, "I'm also using a mirror this year and making it a little bit more severe."

Her comments on dialogue between characters is interesting because it is Cave's ability as a writer of little scenes that stands out in both her book and in her performances. In a short sketch, either with a paper plate or drawn on a page, she has the ability to cut right to the nub of whatever drama is unfolding – exactly as a scene should. As the action depicted is consistent with (and infiltrates) her dialogue and characterisation, it's possible to imagine the whole off-the-page story of the lives she portrays with ease.

If her new book and last year's try-out material is anything to go by, I Loved Her is doubtless one for any comedy fan's Fringe schedule this year.