How do I calculate the unknowns of the current equation for a RLC series circuit?
Now suppose that I want to do the particular conditions for the
  current case to discover A and B. What should I do? I suppose the
  conditions are the same, so, if I have the current equation, I have to
  integrate that to get voltage's


If you have the current equation then A = current at t = 0 and B = di/dt at t = 0. Because all components are in series, the inductor defines the current and, di/dt = V/L as per the well-known equation for an inductor: -

$$V = L\dfrac{di}{dt}$$