How did the English gain control of Dutch and French interests in North America?
The English gained control of Dutch interests much more easily than they did the French interests.
Dutch interests on the continent of North America were limited to the area that is now New York. This area, then called New Netherland, was taken without violence in the 1660s. This happened at a time when Britain and Holland were at war in Europe. The Dutch rulers of New Netherland had sufficiently alienated their people that the people did not try to defend the colony when the English came to attack it.
French interests took much more fighting to conquer. The conquest of the French took place in the French and Indian War, which was part of the worldwide Seven Years War in the 1750s and '60s. The French were pushed out of their Canadian colonies as a result of this war.
The English took control of Dutch lands (New Netherland) in America as a result of the 2nd Anglo-Dutch War in 1664. This area is known today as New York.
The English took control of French lands in America by winning the French and Indian War (the American Theater of the Seven Years War). This was a much greater struggle and caused the English to go into great debt. The Treaty of Paris (1763) gave the English control of French lands from Canada to Louisiana and effectively doubled the English sphere of influence in the New World.