Henry VII became king of England after the Lancastrian forces decisively defeated the Yorkists under Richard III at the Battle of Bosworth Field in 1485?. This victory ended the long running Wars of the Roses between the two houses. His claim to the throne was tenuous and based upon a lineage of illegitimate succession. However, Richard was also a usurper who had little claim to the throne himself.
Principal to Henry's concerns on attaining the monarchy was the question of establishing the strength and supremacy of his rule, since there were a number of potential claimants who had much more obvious claims on the throne. He did this by a number of means, but principally by dividing and undermining the power of the nobility. Other methods include his marriage to Elizabeth of York, an act that unified the warring houses. He was a fiscally prudent monarch who restored the fortunes of an effectively bankrupt exchequer by introducing efficient mechanisms of taxation.
His elder son Arthur having died in 1502, he was succeeded by his second son, Henry VIII. Henry VII did not want the negotiations that had led to the betrothal of his elder son to Catherine of Aragon to go to waste, so he arranged a dispensation for his younger son to marry his brother's widow -- normally a degree of relationship that precludes marriage in the Roman Catholic Church.
Hnery's third child Margaret was married to James IV of Scotland. This marriage lead to James VI inheriting the throne of England as James I after the death of [Queen Elizabeth]?.