Recent experimental studies indicate that polymeric structures with a well-adjusted balance of amphiphilic parts may translocate through self-assembled phospholipid bilayers and enhance the passive trans-membrane transport of smaller molecules. Using a coarse grained lattice Monte Carlo model with explicit solvent we investigate self-assembled lipid bilayers interacting with a linear polymer chain under variation of the hydrophobicity of the chain. Here, we focus on the relationship between the chain's hydrophobicity and its translocation behavior through the membrane as well as induced membrane perturbations. We show, that there is an adsorption transition of the polymer at the bilayer interface, where effectively the solvent phase and the tail phase of the bilayer are equally repulsive for the polymer. Close to this adsorption threshold of the polymer both the translocation probability of the polymer as well as the permeability of the membrane with respect to solvent are enhanced significantly. The frequency of polymer translocation events can be understood quantitatively assuming a simple diffusion along a one-dimensional free energy profile, which is controlled by the effective lipophilicity of the chain and the tail-packing in the bilayer's core.