X-ray surface brightness fluctuations in the core of the Perseus Cluster are analyzed, using deep observations with the Chandra observatory. The amplitude of gas density fluctuations on different scales is measured in a set of radial annuli. It varies from 8 to 12 per cent on scales of ~10-30 kpc within radii of 30-160 kpc from the cluster center and from 9 to 7 per cent on scales of ~20-30 kpc in an outer, 60-220 kpc annulus. Using a statistical linear relation between the observed amplitude of density fluctuations and predicted velocity, the characteristic velocity of gas motions on each scale is calculated. The typical amplitudes of the velocity outside the central 30 kpc region are 90-140 km/s on ~20-30 kpc scales and 70-100 km/s on smaller scales ~7-10 kpc. The velocity power spectrum is consistent with cascade of turbulence and its slope is in a broad agreement with the slope for canonical Kolmogorov turbulence. The gas clumping factor estimated from the power spectrum of the density fluctuations is lower than 7-8 per cent for radii ~30-220 kpc from the center, leading to a density bias of less than 3-4 per cent in the cluster core. Uncertainties of the analysis are examined and discussed. Future measurements of the gas velocities with the Astro-H, Athena and Smart-X observatories will directly measure the gas density-velocity perturbation relation and further reduce systematic uncertainties in these quantities. Show Partial Abstract