Honestly even though I'm sure you could find some estimates, probably the most solid information for that period will come from the 1790 Census, held a couple years later (and mandated by the just-adopted US Constitution). The US Census Bureau has a number of reports and statistics from the 1790 Census available through its website.

Interestingly, the Census information was collected by US marshals (this was practice to 1870), and was collected according to household. Each household was to enumerate the following persons: the name of the head of the family and the number of persons in each household of the following descriptions: the name of the head of household, the number of free white males of 16 years and older, free white males under 16 years, free white females, all other free persons (by sex and color), and slaves. "Indians not taxed" were not counted in the Census.

Some overviews of that data: the United States in 1790 had a still quite small but rapidly growing population of about 3,929,000. About 698,000 of these people (or almost 18%) were slaves.

Heck, since it's only thirteen states we might as well do a table for the population.

State	Total Population	Free Whites	Other Free Persons	Slaves
Connecticut	237,946	232,374	2,808	2,764
Delaware	59,096	46,310	3,899	8,887
Georgia	82,548	52,886	398	29,264
Kentucky	73,677	61,133	114	12,430
Maine	96,540	96,002	538	0
Maryland	319,728	208,649	8,043	103,036
Massachusetts	378,787	373,324	5,463	0
New Hampshire	141,885	141,097	630	158
New Jersey	184,139	169,954	2,762	11,423
New York	340,120	314,142	4,654	21,324
North Carolina	393,751	288,204	4,975	100,572
Pennsylvania	434,373	424,099	6,537	3,737
Rhode Island	68,825	64,470	3,407	948
Tennessee	35,691	31,913	361	3,417
South Carolina	249,073	140,178	1,801	107,094
Vermont	85,539	85,268	255	16
Virginia	747,610	442,117	12,866	292,627

Kentucky and Maine were "Districts" of Virginia and Massachusetts, respectively, and Vermont was in the last stages of settling its claims with New York. Tennessee was "Southwest Territory".

Anyway, that should give a bit of the scope of the state population sizes around 1787. Overall the distance between largest (Virginia) and smallest (Delaware, technically speaking) was not as great as the distance between California and Wyoming today. Slavery had not yet been eliminated north of the Mason-Dixon line, but it can be seen that it was already disproportionately located in southern states, and massively contributed to their population totals (in South Carolina the slave population almost matched the free white population, and it would actually become a supermajority of the state population in the 19th century). Interestingly, Massachusetts was a "big" state in the period - it had been home to some 10% of the total population of the Thirteen Colonies at the time of the Revolution, but its place was steadily slipping (although keep in mind that since the District of Maine was still part of the state, it was the second-biggest state by population in 1790 after Virginia). It was still larger than New York, which actually tended to side with the "small" states at the Constitutional Convention.

Cities themselves were very small in 1790. New York City (ie, the island of Manhattan) had 33,000 inhabitants, and just beat out Philadelphia with 28,000, although that doesn't account for the fact that Northern Liberties was the sixth biggest city (10,000), and Southwark the tenth (5,600), and these have since become neighborhoods in Philly. That should also give a sense how the population was overwhelmingly rural and agricultural in this period (something like 9/10ths of the total).

ETA - whoops, New Jersey got left out. Added in now.