Nucleation free-energy barrier height and size of the critical nucleus are expanded in powers of the chemical potential difference between the supersaturated vapor (or expanded liquid) in the metastable state and the saturated vapor-liquid system in the stable equilibrium state at the same temperature. The coefficients in the expansions are expressed in terms of the thermodynamic properties at the stable equilibrium state. Comparisons with the results obtained from the density-functional calculation for nucleation of the Lennard-Jones fluid show that systematic improvement in predicting properties of the critical nucleus, either liquid droplet or vapor cavity, is achieved by adding the higher order terms in the expansions. The scaling relations proposed by McGraw and Laaksonen are found to be good approximations to the general expansion; in particular, the barrier height displacement appearing in these scaling relations is naturally given as the second order coefficient in the expansion of the barrier height.