What is the weight of a 75kg man 5km from the center of the earth?

To answer the second part of your question first, the weight of a man 1.5km down a mine shaft would be very slightly greater than his weight at the surface (about 10 grams less); because the force of gravity increases in proportion to the inverse square of distance apart (ie, halve the distance, and the attraction would be four times as strong); and he would be slightly closer to the earth's centre of mass..

But the first part of your question is different: Normally, the gravitational attraction of two bodies relates to the distance apart of their centre of masses; not their surfaces. But in the case of a man buried deep INSIDE the earth, the mass above him and around him attracts him in a direction away from the centre of the earth.

At the centre itself, the man would actually feel no gravitational attraction at all (ie, would be weightless), because the attractions from all directions would cancel out.

I would imagine that a man only 5km from the centre of the earth would be very nearly weightless, because most of the attractive forces would cancel out, and only a small attraction towards the centre would remain.