Through history what role did Washington, DC play to become important and influence the US political system, and did it change its influence with time?
Washington DC was basically "made" the capital after Philadelphia after an insurrection by Revolutionary War soldiers called the Pennsylvania Mutiny of 1783. The mutiny showed that state governments could not protect federal institutions so there began a search to move the capital where it could be defended by the Federal government. There was a power struggle between what would be the "Southern" states and the "North" to move the capital somewhere symbolic, and to some in an area they could influence. As part of a deal negotiated by Alexander Hamiliton:
This was after a fight to get the capital either in Philadelphia or New York City, and it was set to move from time to time as part of a deal with the Continental Congress. Yet in Federalist number 43 Madison argues that the capital needs to be on its own land, which eventually won out. Its often been said the city is built on swampland that no one really wanted, and think of the symbolism of that all you want.
It was made to be the capital so it naturally became the power center and dominated US politics by its nature and creation. It's influence has basically been retained over time, though even the Federalists and Anti-Federalists have fought over governments power and reach the capital has been and continues to be the center of politics as it maintains the Federal Bueracracy and Cabinet offices, as well as very close by the Pentagon which is the military headquarters of the US military branches.
Note: I may clean this up later but this should give you an idea of the background.
@MichaelF's answer is a very solid explanation of the origin of DC, and references a historical event that I think deserves more attention. His answer is also far better researched and referenced than I will be able to provide. Having said that, I think the answer to the OP's question ought to include a discussion of the changing role of the national government in US Politics.