Would it be safe to assume that most if not all blacks in the U.S. with deep roots in the country (not including recent migrants from the Caribbeans and other countries) have had ancestors who were actual slaves?
Furthermore, how many generations back would I need to look into to find any link in my family?
That assumption would be safe.
The first black people who showed up in what would become the USA ended up in Virginia on a Portuguese slave ship in 1619. These people were not legally slaves since the colony had no laws about this at that time (that would begin to change by the 1650s). These black people were treated as Indentured Servants and when their term was up, if they had survived, they were freed. Anthony Johnson is an example of this. He married, had children, and grandchildren. His grandchildren were denied access to much of their inheritance due to their race, by their time racial laws had started to be enacted to restrict black people and divide the black, white, and red races.
By the 1650s the plantation colonies really began enforcing genetic slavery, that is, if your mother was a slave, so are you. Imports of black slaves were sold into a culture that had now developed slave laws and manumission was more and less common depending on the region and time period.
I cannot think of any free black people who would choose to migrate to the colonies from Africa, but I would imagine that in the 400 years between the start of slavery and it's end it was not impossible.
But nearly all the black people in the Americas from before the 1960s were descended from slaves. However, since African colonialism ended, more and more Africans have arrived out here.
As for looking back in your family, check everyone around in the 1960s to see where they were born. Then before that, you can probably find genealogy stuff up until the 1860s or so. Before that it gets tricky because slave owners didn't have to keep track of that. Many times the records would be oral records since the slaves tried to keep track of their family as they were sold off to other plantations. After the civil war, many people went looking for family using these networks. Some of these have been preserved, some have been lost.
It gets very hard to track family when you get back to the 1860s and Grandpa James Smith becomes "The big slave James" and the plantation roster lists four James, two Jimmys, and a Jimbo.
Good Luck.