  This paper is concerned with the Hamiltonian elliptic system in dimension
two\begin{equation*}\aligned \left\{ \begin{array}{lll} -\epsilon^2\Delta
u+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -\epsilon^2\Delta v+V(x)v=f(u)\
& \text{in}\quad \mathbb{R}^2, \end{array}\right.\endaligned \end{equation*}
where $V\in C(\mathbb{R}^2)$ has local minimum points, and $f,g\in
C^1(\mathbb{R})$ are assumed to be of exponential growth in the sense of
Trudinger-Moser inequality. When $V$ admits one or several local strict minimum
points, we show the existence and concentration of single-peak and multi-peak
semiclassical states respectively, as well as strong convergence and
exponential decay. In addition, positivity of solutions and uniqueness of local
maximum points of solutions are also studied. Our theorems extend the results
of Ramos and Tavares [Calc. Var. 31 (2008) 1-25], where $f$ and $g$ have
polynomial growth. It seems that it is the first attempt to obtain multi-peak
semiclassical states for Hamiltonian elliptic system with exponential growth.
