The critical behavior of Lennard-Jones (L-J) fluid was investigated by grand ensemble Monte Carlo simulation. Based on the complete scaling, we used Q-parameter method combined with histogram reweighting and finite size scaling theory to estimate critical temperature and density of L-J fluid. The results, Tc*=1.3164(17) and ρc*=0.3168(7), are consistent with former reports. We also calculated the vapor-liquid phase equilibrium properties. The vapor-liquid coexistence curve was obtained by a near-critical scaling algorithm with the help of histogram reweighting technique and hyper parallel tempering method. The critical asymmetry of L-J fluid was investigated. For the L-J fluid we found|t|1-α term dominated the coexistence curve diameter, while the contribution of|t|2β term was weak. |