Typically, the ring structure on mod $p$ Tate cohomology is trivial, in that the only nonzero products involving negative cohomology are those coming from cap products. This is true, for instance, if the center of a $p$-Sylow subgroup of $G$ has rank greater than 1: see this paper of Benson and Carlson. By Chouinard's theorem, this implies that for arbitrary non-periodic groups, all elements in negative mod $p$ Tate cohomology are nilpotent.

The ring structure on mod $p$ Tate cohomology can be split up into 4 parts:

1. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m\geq0$. These just come from products on ordinary cohomology.
2. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m<0$, and $n+m<0$. These are given by cap products when we identify $\hat{H}^m$ with the dual of $H^{-m-1}$.
3. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m<0$, and $n+m\geq 0$.
4. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m<0$.

In fact, for most groups, all products of type 3 and 4 vanish. This is true, for instance, if the center of a $p$-Sylow subgroup of $G$ has rank greater than 1: see this paper of Benson and Carlson (they also show that all products of type 3 vanish iff all products of type 4 vanish). By Chouinard's theorem, this implies that for arbitrary non-periodic groups, all elements in negative mod $p$ Tate cohomology are nilpotent.