Claim to the Throne
Henry Tudor's claim to the English throne was tenuous at best. Through his mother, Margaret Beaufort, he was descended from John of Gaunt's illicit affair with Catherine Swynford.<ref name="Britannia">http://www.britannia.com/history/monarchs/mon40.html</ref> Though the issue born before their marriage had been legitimated by parliament, all claims of this line were never valid until the direct male line of John of Gaunt had become extinct.<ref name="Luminarium">http://www.luminarium.org/encyclopedia/henry7.htm</ref> Henry's paternal grandfather, Owen Tudor, had married Catherine of Valois, the widow of Henry V and daughter to Charles VI of France. Their son Edmund, being the half brother of Henry VI, was created Earl of Richmond. He married Margaret Beaufort, only daughter of John, Duke of Somerset, and died more than two months before their son, Henry Tudor, was born in 1457.<ref name="Luminarium" />
Despite his Lancastrian connections, Henry ultimately ganied the throne through personal battle.<ref name="Britannia" /> At the Battle of Bosworth Field on August 22, 1485, Henry, Earl of Richmond, defeated King Richard III of England and became King Henry VII of England.<ref>http://www.luminarium.org/encyclopedia/bosworth.htm</ref> He was crowned at Westminster on October 30, 1485.
Then, on January 18, 1486 at Westminster, in fulfilment of pledges by which he had gained the loyalty of many Yorkist supporters, he was married to Elizabeth of York, eldest daughter and heiress of Edward IV. Thus the Red Rose of Lancaster and the White Rose of York were united and the pretexts for civil war were done away with-- the Cousins' War had ended.<ref name="Luminarium" />
*Helpful family tree: https://en.wikipedia.org/wiki/Henry_VII_of_England#Ancestry
(note: more information about other parts of his bio and reign at Luminarium and Britannia)