Investigating the physical mechanisms driving the dynamical evolution of young star clusters is fundamental to our understanding of the star formation process and the properties of the Galactic field stars. The young ( 2 Myr) and partially embedded cluster Chamaeleon I is one of the closest laboratories for the study of the early stages of star cluster dynamics in a low-density environment. The aim of this work is to study the structural and kinematical properties of this cluster combining parameters from the high-resolution spectroscopic observations of the Gaia-ESO Survey with data from the literature. Our main result is the evidence of a large discrepancy between the velocity dispersion (σstars = 1.14 ± 0.35 km s-1) of the stellar population and the dispersion of the pre-stellar cores ( 0.3 km s-1) derived from submillimeter observations. The origin of this discrepancy, which has been observed in other young star clusters, is not clear. It has been suggested that it may be due to either the effect of the magnetic field on the protostars and the filaments or to the dynamical evolution of stars driven by two-body interactions. Furthermore, the analysis of the kinematic properties of the stellar population puts in evidence a significant velocity shift ( 1 km s-1) between the two subclusters located around the north and south main clouds of the cluster. This result further supports a scenario where clusters form from the evolution of multiple substructures rather than from a monolithic collapse. Using three independent spectroscopic indicators (the gravity indicator γ, the equivalent width of the Li line at 6708 Å, and the Hα 10% width), we performed a new membership selection. We found six new cluster members all located in the outer region of the cluster, proving that Chamaeleon I is probably more extended than previously thought. Starting from the positions and masses of the cluster members, we derived the level of substructure Q, the surface density Σ, and the level of mass segregation ΛMSR of the cluster. The comparison between these structural properties and the results of N-body simulations suggests that the cluster formed in a low-density environment, in virial equilibrium or a supervirial state, and highly substructured. This work is one of the last ones carried out with the help and support of our friend and colleague Francesco Palla, who passed away on 26 January 2016.Full Tables 1 and 2 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/601/A97Based on observations made with the ESO/VLT, at Paranal Observatory, under program 188.B-3002 (The Gaia-ESO Public Spectroscopic Survey).