1. He had a valid claim to the throne - On his mother's side he was a descendent of John of Gaunt who was the brother of Edward III. His grandmother on his father's side had been married to Henry V. It was not a strong claim however both his father and uncles had been half brothers to Henry VI and had been made Earls. Due to a number of deaths as a result of the Wars of the Roses Henry became the main Lancastrian claim to the throne.
2. Henry was kept safe in exile - Henry left England with his uncle. He spent 14 years in Brittany and despite various attempts to secure his safety if he returned this was never achieved.
3. Richard III's reputation - There was no evidence that Henry had attempted to challenge Edward VIs rule but with his unexpected death his brother Richard of Gloucester seized the throne. Richard's reputation was damaged by the mysterious disappearance of the princes into the tower. Richard faced discontent in the country and a rebellion by the Duke of Buckingham which although unsuccessful weakened Richard's authority. Henry Tudor used this opportunity, initially to support Buckingham, to launch his own attempt to become King of England.
4. Henry attracted support - Henry spent time in Paris and gathered around him men who had disliked Richard's reign. He also attracted experienced and influential men to his cause including the Earl of Oxford, a loyal Lancastrian and Jasper Tudor, both experienced soldiers. Henry landed in Wales and quickly gained support of the Welsh. Most crucially at the Battle of Boswoth Henry was outnumbered and made a direct plea to the Stanley's to join the battle in support of Henry. There entry into the fray was decisive and took Richard by surprise.