Suicide casts a long shadow backwards. Eleanor (Tussy) Marx was the spirited daughter of the father of communism and herself an audacious thinker who spliced feminism with socialism. Hard-working, prodigiously clever and much-loved, she seemed vitality itself. Yet, at the age of 43, she sent her maid out for prussic acid, dressed herself in her white summer dress (it was a cold March day) and lay down to die. That final admission of defeat redefines the woman who seemed so bright and dauntless. Perhaps that's why Holmes has opted for a conventional chronological structure for her biography. To start, as I've done here, with Eleanor's death, is not to spoil an ending so much as to call into question all that has preceded it.

Holmes's prose is energetic and inventive. Her book begins with a bang. "Eleanor Marx changed the world", she declares. Having fired her starting pistol, Holmes is off on a bravura display of verbal gymnastics, scoring full marks for vigour and presentation, if dropping a point here and there on syntactical precision. Readers may want to argue with the author – she will sometimes harass them with an excess of information – but it's clear from the off she will do so with éclat.

Holmes loves and admires the Marx family. Karl is "one of the greatest minds of the age" and Das Kapital "the most influential piece of writing since the Bible, Qur'an, Talmud and the works of Shakespeare". These aren't extravagant claims, but they are made here with engaging warmth and vehemence. Marx's wife, Jenny, is "breathtakingly progressive" and "strikingly lovely". Housekeeper Helene Demuth (Lenchen) is ever kind, ever loyal and a baker of delicious spiced biscuits. The elder girls are serious and graceful. Tussy, black-eyed and ardent, is a prodigy, fluent in several languages, an imperious youngest child. At eight, she is a passionate supporter of "those brave little fellows", the Polish rebels. At nine, she is corresponding with Abraham Lincoln: "I felt absolutely convinced," she wrote later, "that he badly needed my advice." At 13, she is a Fenian. In her mother's words, she was, from infancy, "eine Politikerin von top to bottom".

The Marxes read Shakespeare together, went for long walks and – once they'd moved from the cigar-stinking two-room flat in Soho into a semi-detached in Hampstead – entertained shabbily dressed exiled revolutionaries at exuberant Sunday afternoon at-homes, with fighting talk in a Babel of languages and hearty stews. Tussy stopped going to school at 15, but her education continued, thanks to her "two fathers" – Marx and the aptly named Engels, whose angelic generosity extended to teaching Marx's daughters to know and love ancient literatures from the Norse to the Greek.

Not all was well, though. When she was 18, Tussy left home, rejecting the role that otherwise inevitably awaited her, that of amanuensis to her father. She moved to Brighton, took a teaching job, and persisted, in defiance of parental discouragement, in seeing a lover twice her age – Hippolyte Lissagaray, hero and chronicler of the Paris Commune. Within months, though, she was home again, chastened and anorexic, meekly agreeing to give up her man. She aspired to be an actress, but didn't have the talent. It wasn't until her parents died (within two years of each other, along with her eldest sister and a nephew) that she was free to strike out, aged 28, on "a line of her own".

Holmes is full of enthusiasm for Eleanor's "line" – that of polemicist and political activist – but until Engels died, leaving her enough of his fortune to make her financially independent, Eleanor was a literary drudge. A high proportion of her written work sees her in an ancillary role, as researcher, translator, editor, typist even, of other writers' work. She was a generous teacher. Will Thorne, the trade union leader, who had begun work as a six-year-old, expressed his gratitude at her funeral for the way she had tutored him, giving him the education of which he had been deprived, empowering another rather than making a mark herself. Even the book with which, in Holmes's view, she changed the world, The Woman Question, was co-authored with her lover Edward Aveling. He acknowledged that most of the book's argument was hers, but Tussy was never enough of a bourgeois individualist to claim sole credit.

She was at her most vigorous and effective as an orator. Once, after delivering 40 speeches in three months, she announced her throat was too sore for further speaking, but she was back on the platform within weeks. A slim, passionately committed young woman, she faced down hostile demonstrators and braved police aggression. A cosmopolitan by birth, she was a moving spirit of the Communist International. Her feminist-socialist argument was cogently made. Child labour and sexual discrimination must end – not just for women and children's sake, but because if women were paid less than male workers, then men's wages would always be undercut by cheaper female labour, or by children. Hammering the point home, year after year, Eleanor became one of the most popular and influential figures in the socialist movement.

So why did she put an end to this busy and useful life? Most of her friends and comrades blamed Aveling. No one liked him much. George Bernard Shaw wrote that he had "the face and eyes of a lizard, and a voice like a euphonium". Other comrades found him "repulsive", or like "a low comedian". He sponged off Eleanor, and most of their friends, and fiddled his expenses. He was a shameless philanderer. He exploited and neglected Eleanor – always there when he needed her or her money, never around when she needed him. Already married before he met her, he omitted to inform her when his wife died. Before she poisoned herself, Eleanor discovered that he had secretly married a 22-year-old actress.

Perhaps this last betrayal was all that was needed to reduce her to despair. But Holmes gives almost equal weight to another one.

There is a story, which many but not all scholars accept, that just before Engels died he told Eleanor that her father had had a son with Lenchen, the Marxes' unpaid housekeeper. Eleanor knew the boy, Freddy Demuth, and had believed him to be Engels's child.

Holmes is unable to tell us how Eleanor came to terms with this shock. She had written that her parents were "lifelong friends and lovers" who were "faithful till death". How to square that vision with an adulterous liaison between Karl Marx and the other woman in their household? And, more importantly, how to excuse the deception maintained for decades? How to justify the casting off of Freddy, who was fostered out, given no education or financial help, and made his living as a lathe turner? Holmes has no answer to these questions. But she is surely right that, if Eleanor believed that Marx had fathered Freddy, the news must have turned her world upside down, revealing such nasty things on its underside, that it drastically weakened her will to live. She would have been driven to revise her understanding of her parents' lives, rather as her suicide obliges us to revise, in retrospect, our assessment of hers.

Lucy Hughes-Hallett is the author of The Pike: Gabriele d'Annunzio, which has won the Samuel Johnson prize, the Costa biography award, the Duff Cooper prize and the Paddy Power political biography of the year.