The capture of a straight, infinitely long cosmic string by a rotating black hole with rotation parameter a is considered. We assume that a string is moving with velocity v and that initially the string is parallel to the axis of rotation of the black hole and has the impact parameter b. The string can be either scattered or captured by the black hole. We demonstrate that there exists a critical value of the impact parameter b_c(v, a) which separates these two regimes. Using numerical simulations we obtain the critical impact parameter curve for different values of the rotation parameter a. We show that for the prograde motion of the string this curve lies below the curve for the retrograde motion. Moreover, for ultrarelativistic strings moving in the prograde direction and nearly extremal black holes the critical impact parameter curve is found to be a multiply valued function of v. We obtain real time profiles of the scattered strings in the regime close to the critical. We also study critical scattering and capture of strings by the rotating black hole in the relativistic and ultrarelativistic regimes and especially such relativistic effects as coil formation and wrapping effect.