We discuss fundamental aspects of laser-induced ultrafast demagnetization probed by the time-resolved magneto-optical Kerr effect (MOKE). Studying thin Fe films on MgO substrate in the absence of electronic transport, we demonstrate how to disentangle pump-induced variations of magnetization and magneto-optical coefficients. We provide a mathematical formalism for retrieving genuine laser-induced magnetization dynamics and discuss its applicability in real experimental situations. We further stress the importance of temporal resolution achieved in the experiments and argue that measurements of both time-resolved MOKE rotation and ellipticity are needed for the correct assessment of magnetization dynamics on sub-picosecond timescales. The framework developed here sheds light onto the details of the time-resolved MOKE technique and contributes to the understanding of the interplay between ultrafast laser-induced optical and magnetic effects.