Evolution of the Charge Density Wave state in Cu$_x$TiSe$_2$*

1T-TiSe$_2$ is a quasi two-dimensional CDW material showing a 2x2x2 superlattice modulation below 200 K. Upon Cu doping the CDW is suppressed and superconductivity is induced. We present scanning tunneling microscopy and spectroscopy measurements of the charge-density wave state in 1T-TiSe$_2$, Cu$_{0.05}$TiSe$_2$ and Cu$_{0.06}$TiSe$_2$ single crystals. Topography images at 4.2 K reveal that the charge density waves are present in all samples studied, although the amplitude of the charge modulation decreases with the Cu-doping. Moreover, the chiral phase of the charge density wave is preserved also in Cu-doped samples. Tunneling spectroscopy shows that there is only a partial gap in the pure compound, with bands crossing the Fermi surface. In the Cu-doped samples the system becomes more metallic due to the increase of the chemical potential.