How did Aristotle define life using his elemental system?
Because it seems that Earth, Fire, Air, and Water, only defined non-living matter, and Quintessence was solely used as the element of the heavens...
Or am I wrong and Earth, Fire, Air, and Water were also used to define life?
In De Anima, Aristotle gives a thorough description of his vision of life.
First think to note to answer your question is that Aristotle does not make a distinction between the body and the soul we know today:
the affections of soul are inseparable from the material substratum of animal life
The question you're asking did not really occur to Aristotle, but he does address it:
and things are formed out of the principles or elements, so that soul must be so too
But he then gives a state-of-the-art of soul research and come to the following (terribly funny, if you want my opinion) conclusion:
Each of the elements has thus found its partisan, except earth-earth has found no supporter unless we count as such those who have declared soul to be, or to be compounded of, all the elements. All, then, it may be said, characterize the soul by three marks, Movement, Sensation, Incorporeality, and each of these is traced back to the first principles.
So, to answer your question: souls are, to Aristotle, made up of about 3 of the 4 elements. It doesn't seem much of an issue to him, but he does mention it.
If you want to read more about this, there are very readable translations of De Anima available out there.
