We present an optimal boundary control problem for a Cahn-Hilliard-Navier-Stokes (CHNS) system in a two-dimensional bounded domain. The CHNS system consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn-Hilliard equation for the relative concentration of the fluid. An optimal control problem is formulated as the minimization of a cost functional subject to the controlled CHNS system where the control acts on the boundary of the Navier-Stokes equations. We first study that there exists an optimal boundary control. Then we establish that the control-to-state operator is Frechet differentiable and derive first order necessary optimality conditions in terms of a variational inequality involving the adjoint system.