During an endothermic phase change, what happens to the potential energy and the kinetic energy?
Nothing, if held constant. The interaction would imply an increase in potential energy and either an increase or decrease in kinetic energy, depending on the direction of the phase change.
Those are three different ways of indicating a particular energy state of an object. They do not necessarily affect each other.
"Potential Energy" is a relative term indicating a possible release of energy to the surroundings. If we take its baseline as the overall energy state of a compound, then an endothermic phase change would imply an increase in "potential" as energy is being added to the compound by the system.
A phase change will exhibit an increase in the kinetic energy whenever the compound is changing from a more dense to a less dense phase. The kinetic energy will decrease whenever the compound is changing from a less dense to a more dense phase.