Some brackets would be nice. Assuming you meant f(x) = (x^3 + kx - 4)/(x - 1), you are correct that you need a factor of (x - 1) in the numerator.
Now remember that by the remainder and factor theorems, that for (x - a) to be a factor of a polynomial P(x), then P(a) = 0. Since you need x - 1 to be a factor, substitute x = 1 into the numerator and equate it to 0. You should be able to find k then.