You start by writing down the probability to find a particle at $y$ at time $t$ when it was at $x$ at time $0$, denoted as $K(y,t;x,0)$. You get this by solving the <PERSON> equation with the initial condition $\psi(y,0) = \delta(y-x)$. Then, $K(y,t;x,0) = \psi(y,t)$. Thus, to solve this, we need to know the time development of $\psi(y,t0$.
Let us start with the simple example of a free particle. This is easiest solved in momentum-representation, obtained by Fourier-transforming $\psi(y,t)$:
$$\psi(y,t) = \frac{1}{\sqrt{2\pi\hbar}} \int dp exp(ipy/\hbar) \tilde \psi(p,t)$$ For $\tilde \psi$, the <PERSON> equation gives $$\tilde \psi(p,t) = \frac{1}{\sqrt{2\pi\hbar}} \exp \left(-\frac{i}{\hbar} \left[\frac{p^2 t}{2m} - px\right]\right)$$ This can be inserted back into the equation for $\psi(y,t)$. 526
The integral over $p$ can be solved exactly. The final result is $$K_\text{free}(y,t;x,0) = \sqrt{\frac{m}{2\pi i\hbar t}} \exp\left(\frac{im(x-y)^2}{2\hbar t}\right)$$
Next step: The solution of the <PERSON> equation can generally be written as $$|\psi, t\rangle = \exp\left(-\frac{iHt}{\hbar}\right) |\psi,0\rangle$$ with $H$ being the Hamiltonian of your system. Writing $H = T+V$, the general formula for $K$ becomes $$K(y,t;x,0) = \langle y \mid \exp(-\frac{i(T+V)t}{\hbar}) \mid x \rangle$$ We use the Trotter-Kato Formula (which holds under certain conditions which I won't go into detail at this point. It allows us to write $$K(y,t;x,0) = \lim_{N\rightarrow \infty} \langle y \mid \left[ \exp(-\frac{iTt}{N\hbar}) \exp(-\frac{iVt}{N\hbar})\right]^N \mid x\rangle$$ We insert the unity operator, decomposed as $1 = \int dx | x \rangle \langle x |$ $N-1$ times, which gives us $$K(y,t;x,0) = \int dx_1 dx_2 \dots dx_{N-1} \prod_{j=0}{N-1} \langle x_{j+1} \mid \exp(-iTt/N\hbar) \exp(-iVt/N\hbar) \mid x_j \rangle$$ Note that $V$ as an operator acting on $|x\rangle$ gives just $V(x) |x\rangle$. And $\langle x_{j+1} | \exp(-iTt/N\hbar) | x_j \rangle$ gives us just the contribution of a free particle, i.e. $$\sqrt{\frac{mN}{2\pi i\hbar t}} \exp(\frac{imN}{2\hbar t}(x_{j+1} - x_j)^2$$. 955