Marpa was not a monk. He was a landowner with a wife, Dagmema, and children, farming in southern Tibet, and he taught from that position for his whole life. The Kagyu lineage therefore begins outside the monastery, in a household, and its early figures are conspicuously lay: a farmer, then a hermit in a cave. Monastic institutions came later and were built onto a transmission that had not required them.
The episode everyone knows is the treatment of Milarepa, who arrived carrying the guilt of having killed people by sorcery. Marpa refused to teach him and instead set him to build a tower, then to demolish it and return every stone, then to build another, repeatedly, for years, beating him and driving him away when he asked for instruction. Dagmema intervened on his behalf and was deceived in turn. The tradition reads the whole ordeal as a deliberate purification of the karma of murder, undertaken by a teacher who knew what he was doing and could not explain it at the time.
His own family line ended badly. His son Darma Dode, whom he had intended as heir, died after a fall from a horse, and the transmission passed to disciples instead. Tibetan Buddhism's habit of carrying lineage through students rather than sons is not universal, and here it was forced.