I have purchased an After Effects template which has composition settings of 3840x2160 / 1920x1080 & 1280x720.
I need to output my video at 768x512 which is not in the same proportions to the above settings.
What is the best way for me to alter this project so it outputs at 768x512? I've added it as a new composition but it crops the project and I'm not sure if this is the correct way to do it.
768 X 512 is a very odd size. It is not compatible with many formats. Are you sure that is the size you want? Video has standards and unless you are absolutely sure you need and can use a non standard video size you should reconsider.
If you absolutely need that size then just create a comp the size you want and then drag your original comp into the new comp and scale it uniformly to fit. In many cases will involve some cropping or some letter boxing. You should not ever fit horizontally and vertically because this will distort the image and make circles into ovals and squares into rectangles.