Gibbs-Duhem integration implies the numerical integration of a Clapeyron equation. To start the numerical integration, an initial coexistence point and a corresponding initial slope of the Clapeyron equation are needed. In order to apply Gibbs-Duhem integration to all kinds of systems at diverse physical conditions, one has to investigate and assess the available methods that can be used to compute these initial values. This publication focuses on vapor-liquid equilibria in binary mixtures comprising chain molecules. The initial coexistence point is either computed with the NV? Gibbs ensemble or with the Np?+test molecule method with overlapping distributions, which is introduced in this publication. Although computationally demanding, the Np?+test molecule method with overlapping distributions is applicable at conditions where the NV? Gibbs ensemble fails. We investigated three methods that can be employed to compute the initial slope of the Clapeyron equation. The Widom method and the overlapping-distributions difference method provide correct values for the initial slope. The difference method does only provide the correct answer in special cases. The possibility to judge the reliability of the results makes the overlapping-distributions difference method the safest route to the initial slope. Gibbs-Duhem integration requires the frequent computation of the slope of the Clapeyron equation. This slope depends on ensemble averages of the composition. A new bias method for efficient sampling of the composition in a semigrand-canonical simulation of chain molecules is presented. This bias method considerably enhances the composition sampling in systems comprising chain molecules of different sizes.