If you want to measure the angle (that is what I read in your question), I would go for rotary potentiometers. They are super cheap, lightweight, and easy to read out with a simple microcontroller.
The control of the servos should work with a PID loop to which you feed your setpoint (desired angle) and the measured angle. By adapting the three constants P, I, and D you can define the "behavior of your movements". To control the motors I would recommend you to use a microcontroller instead of the computer. Let the computer calculate the angles for the robotic arm, pass it to your microcontroller, and the
latter should take care of measuring the angles and moving your servos.
If your question is how you calculate the angles of the robot's arm as a function of the end-effector's position, we need to know how many axis you are using, which length the robotic limbs have etc. However, I would have a look at coordinate transformations. Tell us more about the robot's geometry and I will try to help you :).