Let's take a simple system as an example: a single particle (with mass $m$) in a potential $V(\vec{r})$.
The Hamilton-Jacobi equation of this system is
$$ \frac{1}{2m}(\vec{\nabla}S)^2 + V(\vec{r})= - \frac{\partial S}{\partial t}. \tag{1}$$
Schrödinger's equation for the same system is:
$$ -\frac{\hbar^2}{2m}\vec{\nabla}^2\psi + V(\vec{r})\psi = i\hbar\frac{\partial\psi}{\partial t}. \tag{2}$$
To solve (2) you can make the approach
$$ \psi(\vec{r},t) = A(\vec{r},t) e^{iS(\vec{r},t)/\hbar} \quad
\text{with some slowly varying } A(\vec{r},t) \tag{3}$$
Obviously here $A$ is the amplitude, and $S/\hbar$ is the phase of the wave function $\psi$.
When inserting (3) into (2),
then it can be shown that the Hamilton-Jacobi equation (1)
is approximately true (exactly true in the limit $\hbar \to 0$).
For details see also "How do you get quantum Hamilton-Jacobi equation from Schrödinger equation?".
It is true what you have heard: Schrödinger's equation cannot rigorously be derived from classical physics.
Therefore Schrödinger had to guess it with a great amount of intuition.
As shown above it is actually the other way round:
The Hamilton-Jacobi equation (1) can be derived from Schrödinger's equation (2).
The HJE describes classical trajectories of a particle, but not a wave.
Therefore I don't agree with Wikipedia's statement
"The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave."
However, the solution $S(\vec{r},t)$ of the HJE is closely related to a wave.
The surfaces $S(\vec{r},t)=\text{const}$ are the wave-fronts
(the surfaces of constant phase) of a Schrödinger wave-function $\psi(\vec{r},t)$.
The intuition leading Schrödinger to his equation were these requirements:
- It should be a linear differential equation (in order to produce the experimentally observed superposition phenomenons).
- It should be a second-order differential equation in space (in order to produce the experimentally observed waves).
- It should somehow lead to the classical HJE, at least as an approximation.
