On page 114 of the translation of Serre's "Corps Locaux", he states the following:
My question. What is the explicit identification of $L_q\otimes_{\Lambda}A$ with the finitely supported $A$-valued functions on $G^q$ that he is using to obtain this formula?
Some notations for those who don't have the book handy:
$G$ is a group written multiplicatively.
$A$ is a left $G$-module.
$\Lambda=\mathbb{Z}[G].$
$L_q=\mathbb{Z}[G^{q+1}]$ for $q\geq 0.$
This is indeed a bit confusing. $\mathbb{Z} [G^{q+1}]$ is a free $\mathbb{Z} [G]$-module with the diagonal action of $G$. As a $\mathbb{Z}$-module, it has a basis of elements $(g_0,\ldots,g_q)$ where $g_i \in G$, but then every such element may be written as $$(g_0, g_0 g_1, g_0 g_1 g_2, \ldots, g_0 g_1 g_2\cdots g_q) = g_0 \cdot (1, g_1, g_1g_2, \ldots, g_1g_2\cdots g_q)$$ for some $g_0\in G$. It is easy to see that as a $\mathbb{Z} [G]$-module, $\mathbb{Z} [G^{q+1}]$ has as a basis the elements $$[g_1 | g_2 | \cdots | g_q] = (1, g_1, g_1g_2, \ldots, g_1\,g_2\cdots g_q).$$
What Serre probably says is that every element of $\mathbb{Z} [G^{q+1}] \otimes_{\mathbb{Z} [G]} A$ is a finite sum of elements of the form $$[g_1 | g_2 | \cdots | g_q] \otimes a_{g_1,\ldots,g_q},$$ where almost all $a_{g_1,\ldots,g_q}$ are zero, and his identification just views this sum as a function $(g_1,\ldots,g_q) \mapsto a_{g_1,\ldots,g_q}$.
Normally, one writes $$C_q (G,A) = \{ \text{finite sums }\sum_i a_i \otimes [g_{i_1} | \cdots | g_{i_q}] \mid a_i \in A, g_{i_k} \in G \}$$ (considering $A$ as a right $\mathbb{Z} [G]$-module, just to write the coefficients $a_i$ on the left), and then the differentials are given by $$d (a\otimes [g_1 | \cdots | g_q]) = (a\cdot g_1) \otimes [g_2 | \cdots | g_q] - a\otimes [g_1 g_2 | g_3 | \cdots | g_q] + a\otimes [g_1 | g_2 g_3 | \cdots | g_q] - \cdots + (-1)^q \, a\otimes [g_1 | \cdots | g_{q-1}].$$ This probably looks less confusing than Serre's formula.
The differential above comes from the "bar resolution" (see e.g. Weibel, "An Introduction to Homological Algebra", Chapter 6). Note that Serre does not give much details about homological algebra.