Hamilton wanted a strong central government because he was afraid that the state governments would be (as they had been under the Articles of Confederation) too democratic. He worried that the many -- the mass of poor and middle class people -- would control the government.
This worried Hamilton because he believed that the many really did not know what was best for them. He worried that they would force the governments to pass laws like the stay laws that would harm the country's economy. He felt that the elites needed to be in power so that they would do what was best for the country in the long term, not what was best for themselves in the short term.
So Hamilton wanted a strong central government that was somewhat insulated from popular control. That way, the "best" people would lead and they would act in ways that were best for the nation.
Hamilton is credited with creating the American economy, and the stable funding of government, in the 1790s, and he had a classical and realistic grasp on the economics of the time. I think it really bothered Hamilton to participate in the Revolution, only to witness the train wreck of the Articles of Confederation, and the effects of our $4 million war debt on the economy when there were no means by which the federal government could levy taxes to pay it back.
Hamilton also had some monarchist leanings, and some historians suggest that it wasn't so much the fact that we had a king that made Hamilton into a revolutionary, but the fact we had that king, King George III, and how his policies hurt the colonies. Some say he may have been OK with a King George Washington as a benevolent ruler. I think that's a pretty strong interpretation of Hamilton's beliefs, and is probably a stretch, but the idea of a strong central government was indeed in his blood.