What problems did Henry VII face in 1485?

1. The continuation of the Wars of the Roses - The fifteenth century was an unstable time and many would have seen the Battle of Bosworth as a continuation of the wars which had defined the period.  If Henry could win the crown by battle then so could the next claimant.....Henry had to end this pattern if he was to keep his crown.

2. Rival claimants to the throne - Henry's claim remained weak and although Richard III and the princes were dead, the Yorkists' claim lived on through the de la Pole family - John (Earl of Lincoln) and Suffolk (Earl of Suffolk) had a claim at least as strong as Henry's.  Rival claimants gained support from Margaret of Burgundy. She was Edward IV  mother.  She had married the Duke of Burgundy who controlled the Netherlands and France and were an important trading partner with England. 

3. Crown control -Crown control was not equal across the country. It was stronger in the more popular areas, but looser in the borderlands especially due to the lack of developed local governance, 

4. Poor Finances - The crown's financial strength had been depleted by the wars of the Roses and wars abroad