Summary: In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term \[ \begin{cases} -\epsilon^2 \Delta \psi + \epsilon \vec{b} \cdot \nabla \psi + \psi + V(x) \varphi = f (|\eta|) \varphi \quad \text{in } \mathbb{R}^N, \\ - \epsilon^2 \Delta \varphi - \epsilon \vec{b} \cdot \nabla \varphi + \varphi + V(x) \psi = f(|\eta|)\psi \quad \text{in } \mathbb{R}^N, \end{cases} \] where \(\eta = (\psi, \phi) : \mathbb{R}^N \rightarrow \mathbb{R}^2, \varepsilon\) is a small positive parameter and \(\vec{b}\) is a constant vector. We require that the potential \(V\) only satisfies certain local condition. Combining this with other suitable assumptions on \(f\), we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimum of \(V\), convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly indefinite character of the energy functional.