TL;DR In the procedure you posted you forgot to include particular solution. The homogeneous solution will always evaluate to $0$ when used as a solution to the general differential equation.
Homogeneous solution
The roots of the differential equation
$$v''(t) + 2p v'(t) + \omega_0^2 v(t) = \omega_0^2 V$$
are $q_{1,2} = -p \pm j \sqrt{\omega_0^2 - p^2}$. For these roots there are three types of homogeneous solutions:
$$v_h(t) =
\left\{
\begin{array}{lll}
e^{-pt} (A \cos(\omega t) + B \sin(\omega t)), & \omega_0^2 > p^2 & \qquad\text{(Underdamped response)} \\
e^{-pt} (A + B t), & \omega_0^2 = p^2 & \qquad\text{(Critically damped response)} \\
e^{-pt} (A e^{\omega_1 t} + B e^{-\omega_1 t}), & \omega_0^2 < p^2 & \qquad\text{(Overdamped response)}
\end{array}
\right.
$$
where $A$ and $B$ are unknown constants, $\omega^2 = \omega_0^2 - p^2$ and $\omega_1^2 = -\omega^2$. We will now focus only on the underdamped response, as that is the one you analyze in your question.
Particular solution
The particular solution depends on the input (driver) function and the roots of the differential equation. In your case, the particular solution is
$$v_p(t) = K$$
where $K$ is unknown constant.
Solving for unknown constants
The total solution is a linear combination of homogeneous and particular solutions
$$v(t) = v_h(t) + v_p(t) = e^{-pt} \bigl( A \cos(\omega t) + B \sin(\omega t) \bigr) + K$$
If we use this as a solution to the general differential equation we get
$$\underbrace{v_h''(t) + 2p v_h'(t) + \omega_0^2 v_h(t)}_{\text{always } 0} + \omega_0^2 v_p(t) = \omega_0^2 V$$
from which it follows $K = V$. The other two unknown constants are determined from initial conditions $v(0) = v_0$ and $v'(0) = v_0'$
$$A + K = v_0, \qquad -p A + \omega B = v_0'$$
from which it follows $A = v_0 - V$ and $B = \frac{1}{\omega} (v_0' + p v_0 - p V)$.
Final solution
In your special case, $v_0 = 0$ and $v_0' = 0$, and the final solution is
$$v(t) = - V e^{-pt} \Bigl( \cos(\omega t) + \frac{p}{\omega} \sin(\omega t) \Bigr) + V$$
The above solution can be written in a more compact form
$$\boxed{v(t) = V \Bigl( 1 - \frac{\omega_0}{\omega} e^{-pt} \cos\bigl( \omega t - \arctan \frac{p}{\omega} \bigr) \Bigr) }$$
Response overshoot
Note that underdamped (oscillatory) responses naturally have an overshoot. This means that at some point the voltage on the capacitor will be higher than the input voltage.
After the transient response, voltage on the capacitor settles to the input DC voltage
$$v_f = \lim_{t \to \infty} v(t) = V$$
The response overshoot magnitude can be found as
$$\text{PO} = \frac{v(t_m) - v_f}{v_f} \cdot 100\%$$
where $t_m$ is the time of the response maximum. From $v'(t_m) = 0$ it follows that $t_m = k \pi / \omega$ where $k$ is an odd positive number. The response overshoot magnitude is
$$\boxed{\text{PO} = e^{-k \pi p/\omega} \cdot 100\%}$$
