You can use Macaulay's method to find Slope and deflection at c.
At first Use equation of statics to find reaction forces at A and B .( Vertical force and moment about either A or B).
After that use Maculay method To get Equation of bending Moment for whole beam.Use Relation between bending Moment and deflection . Find the constants using boundary conditions at A and B . Then find Slope and deflection at C using the equation. I have attached an example for help.