In this paper, we study the following Hamiltonian elliptic system with gradient term − ϵ 2 Δ ψ + ϵ b → ⋅ ∇ ψ + ψ + V ( x ) φ = ∑ i = 1 I K i ( x ) | η | p i − 2 φ i n R N , − ϵ 2 Δ φ − ϵ b → ⋅ ∇ φ + φ + V ( x ) ψ = ∑ i = 1 I K i ( x ) | η | p i − 2 ψ i n R N , where η = ( ψ , φ ) : R N → R 2 , V , K i ∈ C ( R N , R ) , ϵ > 0 is a small parameter and b → is a constant vector. Suppose that V is sign-changing and has at least one global minimum, and K i has at least one global maximum. We prove that there are two families of semiclassical solutions, for sufficiently small ϵ , with the least energy, one concentrating on the set of minimal points of V and the other on the set of maximal points of K i . Moreover, the convergence and exponential decay of semiclassical solutions are also explored.