If I've understood well your construction, the complex $\hom_k(A,k)$ is the Hochschild complex of the cochain $k$-algebra $C^\star(X,k)$ of the (discrete) space $X$ with coefficients in the $C^\star(X,k)$-module $k$. The $C^\star(X,k)$-module structure on $k$ is given via the augmentation $C^\star(X,k)\rightarrow k$ induced by the inclusion of the base point in $X$. Therefore

$$H_\star\hom_k(A,k)\cong HH_\star(C^\star(X,k),k).$$

Since $X$ is discrete then $C^\star(X,k)=k\times\stackrel{n}\cdots\times k=k^n$ concentrated in degree $0$, where $n=|X|$ is the number of points. Now the Künneth formula shows that

$$H_0\hom_k(A,k)\cong HH_0(C^\star(X,k),k)\cong k^n,$$

$$H_d\hom_k(A,k)\cong HH_d(C^\star(X,k),k)=0,\quad d\neq 0,$$

Therefore

$$H^0A\cong k^n,$$

$$H^dA=0,\quad d\neq 0.$$

If I've understood well your construction, the complex $\hom_k(A,k)$ is the Hochschild complex of the cochain $k$-algebra $C^\star(X,k)$ of the (discrete) space $X$ with coefficients in the $C^\star(X,k)$-module $k$. The $C^\star(X,k)$-module structure on $k$ is given via the augmentation $C^\star(X,k)\rightarrow k$ induced by the inclusion of the base point in $X$. Therefore

$$H_\star\hom_k(A,k)\cong HH_\star(C^\star(X,k),k).$$

Since $X$ is discrete then $C^\star(X,k)=k\times\stackrel{n}\cdots\times k=k^n$ concentrated in degree $0$, where $n=|X|$ is the number of points. Now the Künneth formula shows that

$$H_0\hom_k(A,k)\cong HH_0(C^\star(X,k),k)\cong k,$$

$$H_d\hom_k(A,k)\cong HH_d(C^\star(X,k),k)=0,\quad d\neq 0,$$

Therefore

$$H^0A\cong k,$$

$$H^dA=0,\quad d\neq 0.$$