The very fact that Henry Tudor became King of England at all is somewhat of a miracle. His claim to the English throne was tenuous at best. His father was Edmund Tudor, a Welshman of Welsh royal lineage, but that was not too important as far as his claim to the English throne went. What was important though was his heritage through his mother, Margaret Beaufort, a descendant of Edward III. This descent from King Edward was through his third son, John of Gaunt. John's third wife, Katherine Swynford had borne him several children as his mistress before he married her. The children born before the marriage were later legitmized, but barred from the succession. Margaret Beaufort was descended from one of the children born before the marriage of John and Katherine.
By 1485 the Wars of the Roses had been raging in England for many years between the Houses of York and Lancaster. The Lancastrian Henry later took for his bride Elizabeth of York thereby uniting the houses.
The real matter was decided on the battlefield, at the Battle of Bosworth Field. It was here that Henry and his forces met with Richard III and Henry won the crown. (see quotation above) It was truly through the defeat of Richard and the 'right of conquest' that Henry claimed the throne. It was solidified however, by his marriage to Elizabeth of York, the eldest child of the late king, Edward IV.
The main problem facing Henry was restoring faith and strength in the monarchy. He also had to deal with other claimants, with some of them having a far stronger claim than his own. To deal with this, Henry strengthened the government and his own power, at the expense of the nobles. Henry also had to deal with a treasury that was nearly bankrupt. The English monarchy had never been one of the wealthiest of Europe and even more so after the War of the Roses. Through his monetary strategy, Henry managed to steadily accumulate wealth during his reign, so that by the time he died, he left a considerable fortune to his son, Henry VIII.
It could be debated whether or not Henry VII was a great king, but he was clearly a successful king. He had several goals that he had accomplished by the end of his reign. He had established a new dynasty after 30 years of struggle, he had strengthened the judicial system as well as the treasury and had successfully denied all the other claimants to his throne. The monarchy that he left to his son was a fairly secure one and most definitely a wealthy one.
Henry had seven children by Elizabeth of York, four of whom survived infancy: Arthur, who died shortly after his marriage to Catherine of Aragon (a point of some importance during "The Divorce"), Henry, Margaret and Mary.