In determining 3/5 of the slave population of a southern state, did the southern states count each Slave one by one? Were tax records utilized? or was there another process by which the southern states could count 3/5 of it's slave population and then total the results with the remainder of the southern population to determine representation and taxation?
From 1790 through 1860, the decennial U.S. Census counted slaves in each American state. The 1860 census of slavery is the most famous, not just because it took place at the doorstep of the American Civil War, but because it kicked off a revolution in cartography.
In 1861, cartographers working for Edwin Hergesheimer, head of the U.S. Coast Survey's mapping division, created what is one of the first recognizably modern infographics in American history. "Map Showing the Distribution of the Slave Population of the Southern States of the United States" was revolutionary and led directly to the landmark Statistical Atlas of the United States 13 years later.
The map of slavery directly influenced Northern thought on slavery at the start of the American Civil War, and it was used in military campaigns as well.
The decennial US census, which counted each person in the US, had to have served for this. The enumeration of slaves was just as necessary for determining Congressional representation (and Federal direct taxes, back when they were done) as the enumeration of free people, and these enumerations were the original base reason for the US census. If you check some obvious Wikipedia articles, they have pointers to US Census Bureau data and to compendia that listed counts of free and slave.