In the restricted three body problem, where you consider two objects orbiting each other, such as the sun and earth, and the motion of a third object that does not affect the movement of the first two, but is affected by their gravity, you can sort of figure out how far/fast from one object you have to be to not be orbiting it anymore.
The picture above is taken from Shane D. Ross' Ph.D. thesis. Depending on the total energy of the third mass, it will never be able to go into the shaded areas. So if you are orbiting Earth, which would be $m_2$ in the Sun-Earth example, or $m_1$ in an Earth-Moon one, there is a minimum energy at which can break out of the first and start orbiting the other body. The transition point is the Lagrangian point $L_1$. At a higher energy, it is possible to break away to infinity from both objects, the transition point corresponding to the Lagrangian point $L_2$.
So depending on a more precise definition of you question, a possible answer is that a satellite beyond the Sun-Earth L1 point is more orbiting the Sun than the Earth. The Sun-Earth L1 point is, according to this, about 1% of the way to the Sun. So that's about 1,500,000 Km. You could of course calculate the corresponding $E_1$ enery and translate that to kinetic energy and velocity.