Firstly, this is a complicated question if we want to be accurate linguistically, even most native chinese speakers dont't really realize and know how to describe this. I will try my best to be accurate and clear.
In "zi" and many other chinese "i", it is actually a close close central(central-back in some occurrences) unrounded vowel (https://en.wikipedia.org/wiki/Close_central_unrounded_vowel), you can find the same vowel in some languages, like Russian and Romanian...
To pronounce it properly, you can start with the front version([i]), and slowly push your tongue back towards the central part of your mouth, the muscule less tighten:
Here is a video to show the position when we pronounce it
Again, this is a vowel that does NOT exist in English, you have to create a new sound not used to
Once you get the i(close central unrounded vowel), you will find easy to tell zi and ze. In ze, the vowel is almost the same in firm(british accent, if you want to be more accurate, it is less opened than "ir", it is a close-mid back unrounded vowel (https://en.wikipedia.org/wiki/Close-mid_back_unrounded_vowel)).
But ATTENTION, the chinese pinyin can be tricky with "i" , in some cases it means the front vowel i (which is e in english), some other cases it is the central one (like in Romanian)