The answer to this question is complex because the westward expansion began early in the Colonial period and continued into the nineteenth century. While this includes several centuries, it also describes the enormity of the North American continent. To be definitive, the question needs to be presented in the context of a specific region and time period.
That said, the American West has long been known for its international character. People from the North American East took advantage of opportunities presented by cheap land and mineral wealth, but so did people from all over the world. J S Holliday appropriately titled his definitive portrait of the California Gold Rush of 1849, The World Rushed In (1981) because that is exactly what happened: when gold was discovered in 1848, much of the rest of the world responded. There were "Americans," of course, but there were Europeans and Asians, as well as people from Australia and New Zealand. The same was true of the inspiration presented by the discovery of the Comstock Lode in 1859, in what would become Nevada. By 1870s, Nevada had more foreign born per capita than any other state in the nation. A letter found in Butte, Montana from an Irishman to a cousin in Ireland advised the potential emigrant, "When you cross the Atlantic, don't stop in America, come directly to Butte." The story is told repeatedly throughout the West.
Similarly, the settlement of the Midwest is famous for the way it attracted European farmers who understood it was possible to go to America and find free or cheap land. Midwestern states are checkerboards of ethnicity, remnants of emigration patterns that brought "Americans" and foreigners to settle the region.
Defining exactly "how much" was native Americans and "how much" was first-generation immigrants depends on the place and time. In general, the answer was that there were a lot of immigrants, but there was room for a lot of "Americans" as well.