Google gives the following paper: Greenlees, Tate cohomology in axiomatic stable homotopy theory. It gives a definition of the Tate construction using Bousfield localization and completion, and has some duality theorems, although I couldn't tell if any of them yield Tate duality as you state it.

I think if $M$ is a complex of abelian groups, the Tate construction $M^{TG}$ is the cofiber of the norm map $N: M_{hG} \to M^{hG}$ from homotopy orbits to homotopy fixed points. Lurie's lecture notes introduce the construction in the special case when $G \cong \mathbb{Z}/2 \mathbb{Z}$ and $M$ is a complex of $\mathbb{F}_2$-vector spaces, and give some properties.

Google gives the following paper: Greenlees, Tate cohomology in axiomatic stable homotopy theory. It gives a definition of the Tate construction using Bousfield localization and completion, and has some duality theorems, although I couldn't tell if any of them yield Tate duality as you state it.

I think if $M$ is a complex of abelian groups, the Tate construction $M^{TG}$ is the cofiber of the norm map $N: M_{hG} \to M^{hG}$ from homotopy orbits to homotopy fixed points. Lurie's lecture notes introduce the construction in the special case when $G \cong \mathbb{Z}/2 \mathbb{Z}$ and $M$ is a complex of $\mathbb{F}_2$-vector spaces, and give some properties.