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 <a href="http://www.ams.org/mathscinet-getitem?mr=1182934">this paper of Benson and Carlson</a>.  By Chouinard's theorem, this implies that for arbitrary non-periodic groups, all elements in negative mod $p$ Tate cohomology are nilpotent.