altitude data?

IIRC, the geographic north pole is the point where sea level is closest to the center of volume of the earth, by a healthy enough margin over equatorial locations to rule out anywhere there were early civilizations. The earth bulges out at the middle, so it's all higher (and therefore further from the center) than the poles.

I think you're trying to ask where the point on the surface is such that the center of the earth is most on that side of the earth. This makes sense for any other point inside the earth, such as the bottom of the Mariana Trench, because you can say that, to get to that point from, say, London, you'd have to go past the center of the earth to get there. But the center of the earth is defined as being the point that's just as much directly under Jerusalem as Milwaukee or Tahiti. In order to get any result at all, you need to find the geometric center, and then find the center of relevant stuff, and then you draw a line from the geometric center through the center of relevant stuff and see where it hits the surface. But you obviously can't use the same center for both things, because then the line doesn't go anywhere.

I was thinking you intended to use the center of mass as the second point, but that would give you different answers over time. Imagine a balloon with a rubber ball in it, flying through the air; there's a center of mass, which is the center of the rubber ball, and there's a center of volume, which is the center of the balloon, and you could pick the point on the surface of the balloon that the ball is most directly under. And that's great, but a moment later the relationship between the two has changed and it's a different point.