Aims. We investigate the global galaxy evolution over  ~12 Gyr (0.05 ≤ z ≤ 4.5), from the far ultraviolet (FUV) luminosity function (LF), luminosity density (LD), and star formation rate density (SFRD), using the VIMOS-VLT Deep Survey (VVDS), a single deep galaxy redshift survey with a well controlled selection function. Methods. We combine the VVDS Deep (17.5 ≤ IAB ≤ 24.0) and Ultra-Deep (23.00 ≤ i'AB ≤ 24.75) redshift surveys, totalizing  ~11 000 galaxies, to estimate the rest-frame FUV LF and LD, using a wide wavelength range of deep photometry (337 < λ < 2310 nm). We extract the dust attenuation of the FUV radiation, embedded in the well-constrained spectral energy distributions. We then derive the dust-corrected SFRD. Results. We find a constant and flat faint-end slope α in the FUV LF at z < 1.7. At z > 1.7, we set α steepening with (1 + z). The absolute magnitude M*FUV steadily brightens in the entire range 0 < z < 4.5, and at z > 2 it is on average brighter than in the literature, while φ∗ is on average smaller. The evolution of our total LD shows a peak at z ≃ 2, clearly present also when considering all sources of uncertainty. The SFRD history peaks as well at z ≃ 2. It first steadily rises by a factor of  ~6 during 2 Gyr (from z = 4.5 to z = 2), and then decreases by a factor of  ~12 during 10 Gyr down to z = 0.05. This peak is mainly produced by a similar peak within the population of galaxies with  −21.5 ≤ MFUV ≤ − 19.5. As times goes by, the total SFRD is dominated by fainter and fainter galaxies. The mean dust attenuation of the global galaxy population rises fast by 1 mag during 2 Gyr from z ≃ 4.5 to z ~ 2, reaches slowly its maximum at z ≃ 1 (AFUV ≃ 2.2 mag), and then decreases by 1.1 mag during 7 Gyr down to z ≃ 0. Conclusions. We have derived the cosmic SFRD history and the total dust amount in galaxies over a continuous period of  ~12 Gyr, using a single homogeneous spectroscopic redshift sample. The presence of a clear peak at z ≃ 2 and a fast rise at z > 2 of the SFRD is compelling for models of galaxy formation. This peak is mainly produced by bright galaxies (L ≳ L*z=2), requiring that significant gas reservoirs still exist at this epoch and are probably replenished by cold accretion and wet mergers, while feedback or quenching processes are not yet strong enough to lower the SF. The dust attenuation maximum is reached  ~2 Gyr after the SFRD peak, implying a contribution from the intermediate-mass stars to the dust production at z < 2.