Here's a neat GIF mapping and charting the 10 largest cities in the United States in every decennial census:
There is one story of remarkable continuity here — New York has been the largest city in the USA for a long time, going back to even before Brooklyn was incorporated into the city limits. But there are also some remarkable shifts in evidence like the rise and fall of Buffalo and Detroit in the Rust Belt.
America's No. 2 city, Los Angeles, obviously didn't exist back in 1790 and wasn't even the largest city in California in the 19th century. But over the course of the 20th century it leapt up the charts to finally secure the second slot in 1990 — a position it now shows no sign of surrendering.
One note about the limitations of this data is that by "city" this map means the municipal boundaries of the city, not the population of the larger metropolitan area. That tends to lead to relatively high rankings for a place like San Antonio, which has very expansive borders, and relatively low rankings for places like Boston and DC, which have geographically small central municipalities. DC and Boston "feel" like bigger cities than San Antonio (more pro sports teams, direct flights to more cities, bigger daily newspaper, worse traffic jams) because they are the centers of larger metro areas. The census didn't keep metro area population data back in the day — the concept didn't exist — so municipality data is the only way to do this exercise, but it's less enlightening than a metro area version might have been.