Two waves with the same wavelength that have a path difference can be represented by two sinusoids with a phase difference between them. The phase difference is a function of the frequency or wavelength and wave propagation speed in addition to the path difference, so you need to specify these along with the amplitude of each wave.
For the equations used to add two sinusoids of the same frequency but with a phase difference look at:
The derivation of the equations are also given there. As you can see its rather involved, and I do not think there is a simple intuitive explanation for the general case.
Re. "Why does the path length difference have to be an integer multiple of the wavelength in order to obtain constructive interference?"
If the path length difference is an integer multiple of the wavelength the two sinusoids would be in phase (zero phase shift between them) and add together in a trivial manner giving total constructive interference.
For example $Asin(2\pi\omega t) + Bsin(2\pi\omega t) = (A+B)sin(2\pi\omega t)$.
That is because $sin(2\pi\omega t +n\lambda)=sin(2\pi\omega t)$ where $\lambda$ is the wavelength and n is an integer.