I'm going to preface this by saying. We don't exactly know the origin of the people living in the 13 colonies in 1776. The best guess we have is from the 1790 US census. Using the Surname of the family we attribute the families origin to the most likely place that Surname would originate.

By far the largest population of immigrants to the 13 colonies were African slaves. Accounting for approximately 750,000 individuals in 1790. If you would like to know more about this group I recommend reading up on the Atlantic slave trade.

The next largest non-English group would be German. Germany during this time period was a group of loosely ruled princedoms. Most Germans immigrated for religious reasons (remember the 30 years war ended in 1648 but that didn't end religious problems) or they immigrated for economic opportunity. The fractured nature of Germany and in fact all of Europe in this period made mercenary work the most profitable enterprise in the German states. Those seeking opportunity outside mercenary work would often negotiate passage to America for an agreement of a term of labor. About 100,000 German's immigrated to the US before 1790. A large number of them directly following the Seven Years War. This late immigration meant the total population of German descent in 1790 was only about 280,000.

The next largest population would be Dutch. In the period from 1640-1680 about 4,000 Dutch immigrated to the area that would be the 13 colonies. I say would be because the Dutch often preceded the English. Early arrival and a healthy economy allowed the Dutch Population to boom to 100,000 by 1790.

The next group would be the French. Many of them Huegonots seeking religious freedom. Many of these individuals fled first to England and then to the 13 colonies making it difficult to track or define whether they immigrated from France or England. In 1790 there were roughly 80,000 persons with French surnames.

This website has more raw Data that can help you picture the population of the US in 1790.

I also recommend you check out the source for the above website.

The Source: A guidebook of American Genealogy

Lots of great facts and everything you could ever want to know to trace ancestry.

I'm combining my answer with a previous answer from another thread some time ago about German emigration to America. I add some more parts on the Scots Irish, a huge immigrant populace.

European immigrants came over for reasons like economic opportunity, and for many, religious freedom. Many belonged to “dissenting” groups, like Presbyterians or Moravians, and had some rough experiences back home to say the least. Presbyterians in Ireland faced legal discrimination, including the fact that their marriages weren’t recognized by law. Ireland also face an economic downturn in the first half of the 18th century. As for opportunity, land was cheap (relatively) and plentiful in the Americas as opposed to Europe.

The Scots Irish came in waves to the Americas, starting from the 1710s, halted briefly during the American War of Independence, and then resumed thereafter. Reports say at least 200, 000 immigrated in total, though some say generously 300,000 in the course of 100 years. Many of these Presbyterian Irish came from lower-middling families or very destitute. Some barely had the money to pay for their fare and had to work as indentured servants. More were slightly better off and were able to settle in the “uninhabited” parts of the frontier, though mainly through squatting as often the Proprietors in Pennsylvania complained that settlers took land without paying for it. I put uninhabited in quotes, because the lands were inhabited with natives usually. The Scots Irish mainly settled in Pennsylvania because of its tolerant atmosphere, but by the middle of the century, around the French and Indian War, they migrated south to Virginia and the Carolinas on the Great Wagon road. Vestiges of these migrations can be seen in the Shenandoah Valley, the Carolina Piedmont, and throughout the Appalachians.

Germans also flooded the Americas. Although I want to point out one little thing. "German" as a nationality did not exist until later in the 19th century, so many immigrants would be from small nation-states of the HRE or Prussia, especially in the 18th century. Many emigrated from the Palatinate to Pennsylvania because of its religious tolerance, that's where you get the "Pennsylvania Dutch" from. They usually belonged to the Moravian or Mennonite churches, which got along fine with the Quakers in Pennsylvania. There are actually quite a few reports in the 1720s-40s sent into the Provincial Council talking about all of the "Dutch" coming in droves. They were actually a little worried that they'd be surrounded by the German-speaking peoples. Later reports talked about how they admired their industriousness and relations were a lot more benevolent.

Migration of the "Dutch" to the Americas slowed down by the mid-18th century, by the beginning of the French and Indian/Seven Years War. Though they were still pretty mobile and followed the English and Ulster Scots down the Great Wagon Road into the Carolinas. Old Salem in NC was founded by Moravians. After the AWI, Hessians also settled in.

We have the 1790 census, but I'm careful about using that data as an absolute truth because it leaves out over 80s years of migration and ethnic blending. A lot of scholars estimate that at least a third of the population in mid-18th century America was German, especially in areas like Pennsylvania. Penn seemed to be split in rough thirds with English, Scots-Irish, and Germans, with other groups taking up whatever was left of the pie.

I’d be happy to answer any questions for clarification.

Sources: Sally Schwartz, "A Mixed Multitude": The Struggle for Toleration in Colonial Pennsylvania. Minutes of the Provincial Council of Pennsylvania Vols. 3-5.

We don't have very accurate figures for immigration pre-1790. We don’t have accurate population figures for 1776. The first census was in 1790.

Here are figures for the ethnic origin of the population in the USA in 1790 (assuming that everyone had married inside their ethnic group and there had been no blending – which, of course, was not the case).

Source: I am using the same source as referenced by u/iwinagin, estimation from the 1790 US census.

By far the largest ethnic group would have been the English, with 2,100,000 people. Second would have been blacks from Africa with 757,000. Third would have been Scots-Irish, with 300,000 (These were Ulstermen, there were very few Irish immigrants from southern Ireland until later in the 1800s) . Fourth would have been Germans with 270,000. Fifth would have been Scots with 150,000. Sixth would have been Dutch with 100,000. Seventh would have been French with 15,000. Eighth would have been Welsh with 10,000. All others (other countries or difficult to categorize, but probably mostly English) would have been 204,000.

If we estimate back from this estimation to 1776, we would find a higher proportion of English (50,000-100,000 Tories, mostly of English descent, emigrated from the USA during or after the revolution). A somewhat lower proportion of African descent (importation of slaves continued as well as natural increase).

Contrary to the impression of some, the German ethnic group was not a very large percentage of the population. Under this convention that all ethnic immigration had maintained strict ethnic descent, they would have been about 7% of the population, and this did not increase dramatically. By 1860, under the same convention, German ethnic descent population would have been about 10% of the population.