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You can think about category $Rep(K)$ as about $G-$equivariant sheaves on $G'$. This translates into the following: let $A$ be the algebra of functions on $G'$; this is commutative algebra in the category $Rep(G')$. Using fully faithful functor $Rep(G')\to Rep(G)$ we can consider $A$ as a commutative algebra in category $Rep(G)$. Now $Rep(K)$ is equivalent to $A-$modules in the category $Rep(G)$ as a tensor category. Equivalently, $Rep(K)$ is de-equivariantization of $Rep(G)$ with respect to $Rep(G')$.