The gravitational force of the Earth does not weaken as you go deeper into the Earth. In fact, it increases until you reach the Earth's core, where it is slightly stronger than at the surface. This is because the Earth is not a perfect sphere, but rather an oblate spheroid, meaning that it is slightly wider at the equator than it is at the poles. This means that the Earth's mass is not evenly distributed, and there is more mass closer to the equator than there is closer to the poles. As you go deeper into the Earth, you are getting closer to the center of mass, where the gravitational force is strongest.
However, the gravitational force at the center of the Earth is zero. This is because the gravitational force pulls in all directions equally, so the forces from all sides cancel each other out. This is similar to how an object in orbit around the Earth is not affected by gravity, even though it is constantly falling towards the Earth. This is because the centrifugal force from the object's orbit cancels out the gravitational force from the Earth.
So, to answer your question, there is no point on Earth where gravity is weakest. The gravitational force is strongest at the Earth's core, but it is zero at the center of the Earth.
If you were to go deeper, there would be less of this really dense material (the outer and inner core) actually affecting the gravity you feel, so gravity should then get weaker also. Not only would we have less space on Earth, but our sun would deplete its nuclear fuel much more rapidly meaning that evolution, and life itself, would be greatly curtailed. If gravity were weaker, Earth would be gigantic, and it might be oddly shaped and less mass in the Indian Ocean, making the gravitational pull weaker than any other place on the planet. The Earth's gravitational pull is weakest somewhere around the middle of the Indian Ocean and that's because there is a giant “hole” on that spot.When you dig deeper, the mass above you increases and so does the upper component of gravity. If you assume the Earth composed of infinite spherical shells, you can prove the linear relationship between gravity and distance r