The star formation rate density (SFRD) evolution presents an area of great interest in the studies of galaxy evolution and reionization. The current constraints of SFRD at z>5 are based on the rest-frame UV luminosity functions with the data from photometric surveys. The VIMOS UltraDeep Survey (VUDS) was designed to observe galaxies at redshifts up to ~6 and opened a window for measuring SFRD at z>5 from a spectroscopic sample with a well-controlled selection function. We establish a robust statistical description of the star-forming galaxy population at the end of cosmic HI reionization (5.0<=z<=6.6) from a large sample of 49 galaxies with spectroscopically confirmed redshifts. We determine the rest-frame UV and Ly{alpha} luminosity functions and use them to calculate SFRD at the median redshift of our sample z=5.6. We selected a sample of galaxies at 5.0<=z_spec_<=6.6 from the VUDS. We cleaned our sample from low redshift interlopers using ancillary photometric data. We identified galaxies with Ly{alpha} either in absorption or in emission, at variance with most spectroscopic samples in the literature where Ly{alpha} emitters (LAE) dominate. We determined luminosity functions using the 1/V_max_ method. The galaxies in this redshift range exhibit a large range in their properties. A fraction of our sample shows strong Ly{alpha} emission, while another fraction shows Ly{alpha} in absorption. UV-continuum slopes vary with luminosity, with a large dispersion. We find that star-forming galaxies at these redshifts are distributed along the main sequence in the stellar mass vs. SFR plane, described with a slope {alpha}=0.85+/-0.05. We report a flat evolution of the specific SFR compared to lower redshift measurements. We find that the UV luminosity function is best reproduced by a double power law, while a fit with a Schechter function is only marginally inferior. The Ly{alpha} luminosity function is best fitted with a Schechter function. We derive a logSFRD_UV_(M_{sun}_/yr/Mpc^3^)=-1.45^+0.06^_-0.08_ and logSFRD_Ly{alpha}_(M_{sun}_/yr/Mpc^3^)=-1.40^+0.07^_-0.08_. The SFRD derived from the Ly{alpha} luminosity function is in excellent agreement with the UV-derived SFRD after correcting for IGM absorption.Conclusions. Our new SFRD measurements at a mean redshift of z=5.6 are ~0.2dex above the mean SFRD reported in Madau & Dickinson (2014ARA&A..52..415M), but in excellent agreement with results from Bouwens et al. (2015ApJ...803...34B). These measurements confirm the steep decline of the SFRD at z>2. The bright end of the Ly{alpha} luminosity function has a high number density, indicating a significant star formation activity concentrated in the brightest LAE at these redshifts. LAE with equivalent width EW>25{AA} contribute to about 75% of the total UV-derived SFRD. While our analysis favors low dust content in 5.0<z<6.6, uncertainties on the dust extinction correction and associated degeneracy in spectral fitting will remain an issue, when estimating the total SFRD until future surveys extending spectroscopy to the NIR rest-frame spectral domain, such as with JWST.