what was the nature of royal legitimacy at the time of the Tudors?Watch
Because the Tudor period ended the Wars of the Roses, there were a lot of initial contenders to the throne, but Henry VII took care of that by killing them off.
Henry VIII, his son, took 6 wives in an attempt to produce a male heir (which he succeeded in). There were no real challenges to the throne during his reign, although he was an unpopular king. This may have been due to his continuation of Henry VII's policy of executing anyone who seemed too powerful.
Edward VII, his son, reigned with the support of several advisors (he was just 9 when he took the throne, and 16 when he died). This ensured that most powerful people were involved directly in running the country anyhow.
Edward VII tried to make Lady Jane Grey queen, to prevent the country from falling into the hands of his older Catholic sister Mary- however, Jane did not want to take the throne and was executed 9 days after taking it.
A lot of people questioned Mary I's claim to the throne. Mary I was Henry's first child and Henry had divorced her mother, Catherine of Aragon- which many took as indication of her illegitimacy. It did not help that Mary was an unpopular queen, having tried to convert the country back to Catholicism and murdering those she perceived as heretics. Many did not want a female ruling the country also.
Elizabeth I (sister to Mary I and Edward VII)was at first perceived as weak due to her status as a female, but did a pretty good job at ruling the country. The Tudor dynasty ended with her death, as she had not produced an heir ("The Virgin Queen").
Hope this helps. You can find a lot lot more detail online; it's a fairly well documented subject.