More concrete still? It depends on the field $k$. First, since $\mathbf{Q}/\mathbf{Z}$ is abelian, any continuous homomorphism from $\Gamma_k$ to $\mathbf{Q}/\mathbf{Z}$ factors through the topological abelianization $\Gamma_k^{ab}$ (maximal Hausdorff abelian quotient of $\Gamma_k$), so $H^1(k,\mathbf{Q}/\mathbf{Z})=\mathrm{Hom}_{cts}(\Gamma_k^{ab},\mathbf{Q}/\mathbf{Z})$ is the Pontryagin dual of $\Gamma_k^{ab}$, and is therefore a discrete torsion abelian group. But this can be pretty complicated. If $k$ is a finite field, then $\Gamma_k^{ab}=\Gamma_k=\hat{\mathbf{Z}}$ with the Frobenius automorphism $x\mapsto x^{\# k}$ generating a dense subgroup of $\Gamma_k$. In this case $\mathrm{Hom}_{cts}(\hat{\mathbf{Z}},\mathbf{Q}/\mathbf{Z})=\mathbf{Q}/\mathbf{Z}$ via the map sending a homomorphism to its image on Frobenius. If $k$ is a local or global field, then $\Gamma_k^{ab}$ is described by class field theory, but I'm not sure this gives a "concrete" description of its Pontryagin dual. When $k=\mathbf{Q}_p$ with $p$ odd the abelianization of $\Gamma_k$ is the profinite completion of $\mathbf{Q}_p^\times$, which is isomorphic to $\mathbf{Z}_p\times\mu_{p-1}\times\mathbf{Z}$, so, completing gives $\mathbf{Z}_p\times\mu_{p-1}\times\hat{\mathbf{Z}}$, and then the dual is isomorphic to $\mathbf{Q}_p/\mathbf{Z}_p\times\mu_{p-1}\times\mathbf{Q}/\mathbf{Z}$. This can be generalized to finite extensions of $\mathbf{Q}_p$ (you'll get extra copies of $\mathbf{Z}_p$). Probably another user smarter than I can elaborate on this in the case of global fields (or point out any oversights or mistakes on my part). (This isn't really an answer but was too long for a comment.)
EDIT (added to address the second question):
The restriction map $H^1(k,\mathbf{Q}/\mathbf{Z})\rightarrow H^1(K,\mathbf{Q}/\mathbf{Z})$ just sends a homomorphism $G_k\rightarrow\mathbf{Q}/\mathbf{Z}$ to its restriction to $G_K$, so a homomorphism is in the kernel if its restriction to the latter subgroup is zero. When $K$ is Galois, this kernel therefore consists of the continuous homomorphism $G_k\rightarrow\mathbf{Q}/\mathbf{Z}$ that factor through $\mathrm{Gal}(K/k)$, which is $\mathrm{Hom}_{cts}(\mathrm{Gal}(K/k),\mathbf{Q}/\mathbf{Z})=H^1(K/k,\mathbf{Q}/\mathbf{Z})$. This is the inflation-restriction sequence for $K/k$.