Henry VII, known as Henry Tudor, was a great times three grandson of Edward III. He became king on August 22, 1485 after defeating Richard III at the Battle of Bosworth Field.
On Henry Tudor rested the hopes of the House of Lancaster in the last stages of the Wars of the Roses. After he was crowned king, he married Elizabeth of York, one of the surviving daughters of the Yorkish king, Edward IV. The marriage combined the two houses. He even combined the symbols of the York rose and the Lancaster rose into a Tudor rose.
Henry's Lancastrian claim was through his mother, Margaret Beaufort, who was a great granddaughter of the first Duke of Lancaster, John of Gaunt. Margaret Beaufort was descended from John Beaufort, one of John of Gaunt's children by his third wife, Katherine Swynford, who had been governess to his children by his first wife before becoming his mistress. John of Gaunt married Katherine after all their children were born. A papal bull and royal patent legitimized their children, giving them the patronym Beaufort, but explicitly excluded the Beauforts and their heirs from the royal succession. Thus, any claim Henry VII made to the throne through the Beaufort line would be controversial and subject to challenge -- and perhaps more war.
Henry VII's father, Edmund Tudor, died before Henry's birth. Edmund was the son of a Welsh squire, Owen Tudor, who secretly, and without royal permission, married Catherine of Valois, a daughter of Charles VI of France. Catherine of Valois was the widow of Lancastrian King Henry V. Henry's V's father, Henry IV, had deposed his own cousin, King Richard II, who then died mysteriously in prison, so any claim through Henry V would be controversial. But Henry VII's claim was not through a prior marriage of an ancestress; this connection could not justify Henry VII's claim to the throne. The scandal of the secret marriage arguably made his family reputation shakier. It was undeniable that Catherine of Valois gave him royal ancestry, but that was distant, French, and through a daughter of the king, not a son.
Henry Does an End Run
With all this family history of birth and heritage controversies, Henry VII came to power knowing that his own claim to be the rightful king would be controversial. He also knew of the controversies surrounding his predecessors (see links below), and how those controversies had contributed to challenge, conflict and war. Henry VII likely had his eye on stability, for himself and his heirs.
So Henry VII didn't claim the throne through right of inheritance: he claimed it through the right of conquest, not through any of his own royal lineage. And his marriage to Elizabeth of York, which could be seen as making sure his children were of the lineage of Edward IV? He made sure that marriage happened after he'd been crowned, thus making his own claim solely and completely on the right of conquest. Henry VII made sure that any "birther" controversies would be totally irrelevant to his right to the crown, and to the right of his descendants as well.
