Experiments on the heating and melting of two-dimensional finite dust crystals are performed using random laser excitation of the dust particles by a rapidly moving laser beam. The achievable dust temperatures scale with the square of the laser power. The heating process is described for different dust clusters under various plasma and cluster conditions. A single-particle model is developed to explain the observed behavior of the cluster under the random laser excitation. Good quantitative agreement is found when the radiation pressure is made responsible for the particle excitation by the laser. The dynamical properties of the system during heating are analyzed and the dominant modes are identified. From this, it is demonstrated that the heating process is of a nearly equilibrium nature in contrast to previous melting experiments. Finally, the melting of the dust cluster by laser heating is studied. From these experiments, a precise determination of the critical coupling parameter for the solid-fluid transition was possible. It is measured as Gamma = 270-480 for an N = 18 cluster.