I like the previous answer, but I thought you might want something less abstract. To simulate the Schrodinger equation, replace space by a grid of points, and the wavefunction values with a wavefunction value on each point of the grid. Then replace:
$$ \nabla^2 \psi $$
in the Hamiltonian with
$$ (\sum_e \psi(x+e) ) - N \psi(x) $$
Where e runs over the basic lattice steps--- to nearest neighbors in each direction--- and N is the number of nearest neighbors on the lattice, so that in one dimension, you have $\psi(n)$ where n is an integer (a one dimensional grid), and the Laplacian is
$$ \nabla^2\psi(n) = \psi(n+1) + \psi(n-1) - 2\psi(n) $$
In two dimensions you have $\psi(i,j)$ where i,j are integers:
$$ \nabla^2\psi(i,j) = \psi(i+1,j) + \psi(i-1,j) + \psi(i,j+1) + \psi(i,j-1) - 4 \psi(i,j) $$
Then you write down the discrete Schrodinger equation
$$ i \partial_t \psi = \nabla^2 \psi + V(x) \psi $$
With the appropriate discrete form and simulate away. Notice that the dimension of the state space is the number of grid-points, and the dimension of the Hamiltonian matrix is the square of the number of grid points, but these matrix elements are only nonzero for a few positions, so there are only about as many nonzero matrix elements as there are gridpoints. This makes it inefficient to represent H as a matrix in a computer, it is what is called a "sparse matrix" in computer science.
You should also know that there is a stiffness problem with this equation--- if you make the lattice spacing of x $\epsilon$ (this just rescales the Laplacian), the discrete spacing in time for the purposes of simulation needs to be smaller than about $\epsilon^2$ for a naive integration algorithm. You can use a better algorithm (this is called a stiff integrator), but to get qualitative intuition, just use a coarse lattice and a fine time lattice, or solve the time-independent (eigenvalue) problem. I got bitten by the stiffness when I was in your shoes.
You should also know the fact that the discrete form is the actual correct form of the Schrodinger equation describing electron motion in a real atomic lattice, as in a metal. This is usually taught as the "tight binding approximation", and derived from consideration of the continuous Schrodinger equation separating into bands in a periodic potential, but the end result is nothing more than the discrete form of the Schrodinger equation you would write down as above.
The Schrodinger equation is developed from this discrete point of view in The Feynman Lectures on Physics Vol III, which is appropriate for a high school student, since it has no prerequisites.