I would not consider benzene to be an alkene because it's chemical reactivity differs significantly from the reactivity of alkenes. For example, a classic chemical reaction involving alkenes is an addition reaction with elemental bromine. Benzene will not undergo this reaction and many other reactions characteristic of alkenes (e.g. catalytic hydrogenation, epoxidation, etc.).
As noted by queenofbabes, this difference in chemical reactivity is due to the fact that benzen's electronic structure (i.e. it's molecular orbitals) is very much different than the electronic structure of typical alkenes.