This work experimentally investigates the impact of p-doping on the relative intensity noise (RIN) properties and subsequently on the modulation properties of semiconductor quantum dot (QD) lasers epitaxially grown on silicon. Owing to the low threading dislocation density and the p-modulation doped GaAs barrier layer in the active region, the RIN level is found very stable with temperature with a minimum value of -150dB/Hz. The dynamical features extracted from the RIN spectra show that p-doping between zero and 20 holes/dot strongly modifies the modulation properties and gain nonlinearities through increased internal losses in the active region and thereby hinders the maximum achievable bandwidth. Overall, this Letter is important for designing future high-speed and low-noise QD devices integrated in future photonic integrated circuits.