The most important role for the new central government was, in Alexander Hamilton's view, to create an environment in which business could prosper. Hamilton envisioned a country in which merchants, traders, and industrialists would lead the economy. It was the role of the national government to make an environment in which that sort of economic activity could thrive.
It was for this reason that Hamilton set in motion his "American System." Hamilton wanted a central bank, tariffs, and internal improvements. All of these things were meant to make it easier for industry and such to get established and grow in the new country. For Hamilton, the government was not supposed to be a small thing that got out of the way of small farmers. It was, instead, supposed to be an active part of creating a mixed, modern economy in the US.
Being a Federalist, Hamilton envisioned a large role for the new government. Hamilton believed that if the best and the brightest individuals in the new nation were in the position of power in the new government, then its reach should be expansive as it would be considered helping to solve problems in the new nation. As a Federalist, Hamilton strongly believed that the federal aspect of government should be applicable to the entire nation in the form of a strong federal government. His advocacy of a national bank is representative of this. The idea of a federal bank is one where Hamilton showed that national government can solve problems that plague the country, such as the presence of a large debt. This can also be seen in Hamilton's advocacy for the excise tax on whiskey. Hamilton believed in the power of the federal government, and did not shy away from its influence in as many realms as possible in American life. Hamilton's embrace of a flexible approach to the Constitution in order to allow the federal government more power to solve national problems also demonstrates his belief in the enhanced power of the federal government in national affairs.