Group cohomology is sheaf cohomology on a certain site, see e.g. Tamme's book on étale cohomology. The Hochschild-Serre spectral sequence is a Leray spectral sequence. Galois cohomology is étale cohomology of fields.
Using étale cohomology, it is trivial to compute the Brauer group of a local field: $\mathrm{Br}(K) = H^2(K,\mathbf{G}_m)$ and one has an exact sequence $0 \to H^2(\mathfrak{O}_K,\mathbf{G_m}) \to H^2(K,\mathbf{G}_m) \to H^1(k,\mathbf{Q}/\mathbf{Z}) \to 0$. But $H^1(k,\mathbf{Q}/\mathbf{Z}) = \mathbf{Q}/\mathbf{Z}$ and $H^2(\mathfrak{O}_K,\mathbf{G_m}) \hookrightarrow H^2(k,\mathbf{G}_m) = 0$ since $G_k = \hat{\mathbf{Z}}$ and $H^1(k,\mathbf{G}_m) = 0$ (Hilbert 90) and the Herbrand quotient is trivial.