I am using the definition of compactly generated space from The Category of CGWH Spaces, which is
In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in $C$ for any $u: C\to X$ where $C$ is compact Hausdorff.
A space is compactly generated if all $k$-closed subsets are closed.
locally compact means every point has a local base of compact sets.
This is different from the definition from Wikipedia
So, is any compact space compactly generated?
And, is any locally compact space compactly generated?
It seems the following.
We shall use this question by Amathstudent and its answer by Brian M. Scott.
We prove that each weak Hausdorff compactly generated $T_1$ space is $KC$. Since there is a weak Hausdorff compact $T_1$ space, which is not $KC$ (see the space $\Bbb Q^*\times\Bbb Q^*$ in the answer by Brian M. Scott), this space is not compactly generated.
So, let $X$ be a weak Hausdorff compactly generated $T_1$ space and $Y$ be a compact subset of $X$. We claim that $Y$ is $k$-closed subset of $X$. Indeed, let $C$ be a compact Hausdorff space and $u: C\to X$ be a continuous map. Since the space $X$ is weak Hausdorff, the set $u(C)$ is closed in $X$. A set $u(C)\cap Y$ is compact as a closed subspace of a compact space $Y$. By Lemma 1, the space $u(C)$ is Hausdorff. So the set $u(C)\cap Y$ is closed in the space $u(C)$. Since the set $u(C)$ is closed in $X$, the set $u(C)\cap Y$ is closed in the space $X$ too. Since the map $u$ is continuous, the set $u^{-1}(Y)= u^{-1}(Y\cap u(C))$ is closed in $C$. Since the space $X$ is compactly generated, the set $Y$ is closed in $X$. Hence $X$ is a $KC$-space.