Since G is cyclic with order q-1, the image of G consists of units in R that produce 1 when raised to the q-1. Nothing changes if R is restricted to the base ring and the q-1 roots of 1. (The base ring is either Z or Zp for some prime p.) That's all we need to support the character χ(G).
By convention, χ(0) = 0. With this in place, χ respects multiplication in the two rings. This does not mean χ is a ring homomorphism, since it does not respect addition, but it is compatible with multiplication mod q.
Review cyclotomic extensions, and in particular, cyclotomic polynomials. The reduced ring R is a subring of Z[y] or Zp[y], where y is a root of the cyclotomic polynomial ζq-1(x). Remember that ζ is the product of x-y, where y ranges over the primitive q-1st roots of 1.
In addition to the aforementioned roots of 1, adjoin the qth roots of 1. These are the roots of ζq(x), which is xq-1 divided by x-1. The image of G, under χ(), never touches these roots, since q and q-1 are coprime. The qth roots of 1 are brought in to support the gaussian sum.
There are at least two characters into R, the trivial character G → 1, and the quadratic character χ(x) = [x\q], which is the legendre symbol x over q. If R includes other roots of order q-1, more characters are possible.
Let y be a primitive qth root of 1. The gaussian sum τ of a character χ, written τ(χ), is the sum over all j in Zq of χ(j)×yj.
To be fair, this depends on y, and perhaps it should be written τ(χ,y), but a primitive root y is usually fixed, and τ is defined in that context.
Here is a simple example. Let q = 3, and let χ be the quadratic character that maps 2 to -1 and 1 to 1. Note that y-y2 is not the same as y2-y.