We use the position of the sun (or more accurately the centre of mass of the solar system) as this gives us a very nearly inertial frame of reference. An inertial frame is one which is not changing its velocity. The surface of the Earth does not define an inertial frame.
The Earth orbits the sun and so it is moving in opposite directions in summer and winter. If we were to use the radial velocity (for example) relative to the Earth, we would get a different value in summer from winter. The orbital speed of the Earth is about 30km/s, and the radial velocity of stars tends to be in the 10s of kilometers per second, so the Earth's motion is a significant amount. If, however, we use the centre of mass of the solar system (which is always very close to the sun) then we can record the actual motion of the star.
The process of finding the relative motion is to measure the radial motion relative to the telescope doing the observations, and then subtract the velocity of the Earth around the sun, and the velocity of the telescope relative to the centre of the Earth. Since these two velocities are well known, compensating for them is essentially no more than subtraction.