I have followed the GUIDED CYGWIN INSTALL instructions on the site you linked. The mirror they suggest doesn't seem to up anymore. I have tried other mirrors but in the end when I try to run Singular out of the Cygwin window I just get error messages.

Have you used singular? I have been trying to install and I am struggling.

@VladimirDotsenko Yeah good point how do I reflect that in the notation? Am I OK using the isomorphism theorem to say $\mathrm{im} \xi_0\cong A/\mathbb{K}$?

This argument must be false because the same argument could be used for the Weyl algebra which has a trivial centre. But every derivation is inner there, so $HH_1$ would be 0. Again thinking out loud I may very well be mistaken.

Am I thinking of this in the right way? We want to find $$HH_1(A):=\mathrm{ker} \xi_{1}/\mathrm{im} \xi_{0}.$$ We know $$\mathrm{ker} \xi_{0}=Z(a),$$ where $Z(A)$ is the centre of $A$ and thus isomorphic to $\mathbb{K}$. We also know $\mathrm{im} \xi_{0}\cong A/\mathbb{K}$, therefore $$HH_1(A):=\mathrm{ker} \xi_{1}/(A/\mathbb{K}).$$ Does this mean $$HH_1(A)\cong \mathbb{K}?$$

@VladimirDotsenko Thank you for your suggestion I am trying to use it.

@VladimirDotsenko I am using a projective resolution which is formed by taking a total complex of a double complex. I haven't actually written out explicitly the total complex before I take Hom. Would it be useful for me to write down the proj res explicitly?