Solving RC circuits with impulse or unit step input by using convolution integral
-Firstly You should find the impulse response of Vc for given impulses 

differently.

-Then you know that capacitor acts like short circuit in impulse response .

-You find the impulsive current then find voltage by integrating the current in given t so you can find the impulse response.

-Secondly do the same thing for second input.

-For zero state solution you should convolve your first impulse response with first input waveform and second impulse response with second input waveform.

-Convolve  0 to t which you want to find zero state solution then add zero state solution to zero input solution. 

I think this can be helpful.