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We aim at characterizing the accretion properties of several hundred members of the star-forming cluster NGC 2264 (3 Myr). We performed a deep u,g,r,i mapping and a simultaneous u+r monitoring of the region with CFHT/MegaCam in order to directly probe the accretion process from UV excess measurements. Photometric properties and stellar parameters are determined homogeneously for about 750 monitored young objects, spanning the mass range 0.1-2 Mo. About 40% are classical (accreting) T Tauri stars, based on various diagnostics (H_alpha, UV and IR excesses). The remaining non-accreting members define the (photospheric+chromospheric) reference UV emission level over which flux excess is detected and measured. We revise the membership status of cluster members based on UV accretion signatures and report a new population of 50 CTTS candidates. A large range of UV excess is measured for the CTTS population, varying from a few 0.1 to 3 mag. We convert these values to accretion luminosities and obtain mass accretion rates ranging from 1e-10 to 1e-7 Mo/yr. Taking into account a mass-dependent detection threshold for weakly accreting objects, we find a >6sigma correlation between mass accretion rate and stellar mass. A power-law fit, properly accounting for upper limits, yields M_acc $\propto$ M^{1.4+/-0.3}. At any given stellar mass, we find a large spread of accretion rates, extending over about 2 orders of magnitude. The monitoring of the UV excess on a timescale of a couple of weeks indicates that its variability typically amounts to 0.5 dex, much smaller than the observed spread. We suggest that a non-negligible age spread across the cluster may effectively contribute to the observed spread in accretion rates at a given mass. In addition, different accretion mechanisms (like, e.g., short-lived accretion bursts vs. more stable funnel-flow accretion) may be associated to different M_acc regimes., 24 pages, 21 figures, accepted for publication in Astronomy & Astrophysics