The British came to America to rival Spain,France and Holland's colonies. They also wanted to expand their empire and to become wealthy. They are on their way to the Chesapeake Bay to settle a colony after Raleigh's colony failed.
No money for you.
One of the problems that Jamestown had was that Queen Elizabeth would not fund the colonies.
Another problem was mosquitos that spread diseases throughout the colony.
There was a period known as the staving time where there was poor leadership, harsh winters and lack of food.
They found success from growing a cash crop that John Rofle discovered called Tobacco. Tobacco was very valuable to England.
Jamestown also had problems with the Native Americans that kept on attacking them.