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Since the Gulf Stream is warm, a significant amount of water evaporates as it crosses the Atlantic, leaving it very salty. Much saltier than the “normal” water in the Atlantic. So in principle one would expect to find this water along the Atlantic floor.

But the Gulf Stream is also a lot warmer than “normal” Atlantic water. Thus the Gulf Stream is found near the surface of the Atlantic Ocean.

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The Gulf Stream enters the Arctic Ocean, and is rapidly cooled down.
At this point there is no heat to keep it at the surface and it sinks. In fact, this is much like a gigantic waterfall below the ocean surface. This waterfall effect is called the thermohaline circulation (thermo = temperature
haline = salinity).
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In the Atlantic, the current splits in two, the southern part headed for Africa, the other part pulled north by the thermohaline circulation in the Arctic Ocean.