Is a river potential or kinetic energy?

Both, if you refer to a flowing stream. The topographic elevation of the streambed above the base level toward which it leads (e.g., sea level) is indicative of the potential energy of the water in the river. (Another kind of potential energy is proportional to the water depth above the streambed.) The river's kinetic energy is proportional to the square of its current velocity and to the mass of moving water. Remember KE = (m v^2)/2 from physics?

If the river ceases to flow, for example enters a lake or impoundment, its kinetic energy will decrease to zero. Yet because of potential energy, its water may still have considerable energy.