As I understand the construction of the determinant of a perfect complex, this definition is quite straightforward, following from the fact that in a short exact sequence, say
$$ 0\rightarrow S\rightarrow E\rightarrow Q\rightarrow 0$$
defining the determinant of the sequence to be the alternating tensor is the canonical way to make it isomorphic to $\mathbb{1}$.

Also, I think good references to this may be the original paper by Knudsen-Mumford, a book by Kato, and also a paper by kings which are listed below:

Knudsen, Finn Faye; Mumford, David: The projectivity of the moduli space of stable curves. I. Preliminaries on ''det'' and ''Div''. The part about determinants appears in Chapter I, but note that there is a typo defining the determinant, namely in the map of the transposition of tensor product, there should be $\alpha\cdot\beta$ instead the sum of these two as a power of $-1$;

Guido Kings: An introduction to the equivariant Tamagawa number conjecture: the relation to the Birch-Swinnerton-Dyer conjecture

http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/FGAlgZyk/index-en.html

There is a part about determinants in lecture 1 section 5, where there are not a lot of details but it provides a good view towards the construction of determinant.

Kato: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B_dR, part I, which mentions determinant in 2.1.