The 1st guy's right about Richard Lionheart. And yes, that's why the Tudor rose is red & white.
Henry VII and Elizabeth of York were both descended from Edward III, so they were distant cousins. It's a very tangled family tree.
Henry's mother was Margaret Beaufort, who was in turn the granddaughter of John of Gaunt, 3rd son of Edward III, and John's 3rd wife & former mistress Katherine Swynford (Read Katherine by Anya Seton for the story--she's fairly accurate; got the important details down. Great story--but I digress). Henry's father was Edmund Tudor, half-brother of the last Lancastrian king--Henry VI (their shared mother, Katherine d'Valois married 2x--Henry V & a Welshman named Owen Tudor/Owain ap Meredith ap Tudor)
Elizabeth was descended from three of Edward's sons: Lionel, Edmund, and John (through Joan Beaufort); her own father was Edward IV, and he married a relatively minor noblewoman rather than a princess.
Probably the main reason they got married was the time-honored tradition of using a marriage to end a war or seal a treaty. Have the heirs wed and ta-da! you end the war! By marrying Elizabeth, Henry linked his weaker claim to a strong one and (with her brothers missing) she was the senior heir to the Yorkists, not just the most age-appropriate unwed female. Theoretically, the children have the best claim of all. Which pretty much worked in this case.
As for killing off his own relatives, well, he's not the 1st nor the last monarch to do that. Most of the other Plantagenets he or Henry VIII killed had better claims to the throne, so they looked at it as a way of protecting themselves. And given that Henry VII spent much of his youth in exile, I can't think he had much family feeling in that sense. He always struck me as a very ruthless man.
Hope this helps!