Do I need to take into account the internal optical system of the phone to get meaningful results?
A tiny camera like that on a phone is well approximated by an infinity conjugate objective, unless it is focussing on really close objects. So you would design an infinity conjugate of the numerical aperture and color correction appropriate to your application.
So, at the very least, you need to work out the stop diameter from the camera's $f$ number and also the effective focal length.
However, the phone camera is probably not going to be perfectly corrected and its chromatic and wavefront aberration will hinder your ability to get the point spread function down to the pixel size. This may or may not be a problem, depending on your application. If the phone's PSF is too big for your application, then you do indeed need to know exactly what the lens system is so that these aberrations can be corrected in your lens.
You also need to do proper research on the autofocus software to see how the latter will react to an external lens. If the phone has a separate rangefinding infrared sensor, this is going to be grossly miscalibrated by the presence of an outside lens and will wreak havoc on you application. If, on the other hand, the autofocus software works through image processing to optimize sharpness of edges, then it will help you. You need to decide how you're going to interact with the image acquisition software: if it's an external range-finding sensor then you'll need to switch autofocus off and control the camera completely from your own microscope app that you will need to develop.