I don't exactly know, which approach you are looking for. I know at least three. I've written out one in my QFT lecture notes using the method of path integrals. It's one of the few examples where one can calculate the lattice-version of the path integral exactly and then take the continuum limit. It's not the most convenient approach, but it helps to understand path integrals. The first chapter of my QFT lecture notes is about non-relativistic quantum theory to introduce path integrals. Perhaps this helps you:
http://fias.uni-frankfurt.de/~hees/publ/lect.pdf
The other two approaches are:
(a) use the Heisenberg picture of the time evolution, solve for the operator-equations of motion and then evaluate the propagator as
U(t,x;t_0,x_0)=\langle t,x|t_0,x_0 \rangle,
where |t,x \rangle are the time-dependent position eigenvectors in the Heisenberg picture.
(b) use the energy eigen functions and resum the corresponding series for the propagator. This is a quite tricky business using the integral representation of the Hermite polynomials.
You find these two approaches in my German QM manuscript:
http://theory.gsi.de/~vanhees/faq/quant/node18.html
http://theory.gsi.de/~vanhees/faq/quant/node49.html
Perhaps you can follow the calculations even without understanding the German text. There are quite a lot of steps written out in formulas :-).
