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_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.