There isn't much written up on how to compute the higher chiral homology of vertex algebras. Besides the original work of Beilinson and Drinfeld that covers in particular the universal cases of Virasoro and Affine algebras you won't find much more. There's an unwritten theorem by Gaitsgory that proves the vanishing for simple Affine algebras at integral level.

In the case of elliptic curves and in the limit of the nodal elliptic curve, together with Van Ekeren we related the first chiral homology group to the first Hochschild homology of the Zhu algebra (just as comformal blocks are related to the zeroth Hochschild homology) in https://arxiv.org/abs/1804.00017

Unfortunately that relies on a technical condition of vertex algebras being "classically free", that is that their classical limit is isomorphic to the arc algebra of the C2 quotient. This was shown to be the case for minimal models $Vir_{p,p'} if and only if $p=2$. And some simple affine algebras at integral level.

We can now relax this condition to vertex algebras to the case then the classical limit is a quotient of the arc algebra by a finitely generated differential ideal. This is still very hard to prove. The only known example of this to my knowledge is the Ising model $Vir_{3,4}$ in

https://arxiv.org/abs/2005.10769

Using this we can now prove the vanishing of the first chiral homology group of an arbitrary elliptic curve (not necessarily the nodal limit) with coefficients in either the 2,p' minimal models or the Ising model. As well as recover some of Gaitsgory's results. That article should come out soon, it's being finalized for a while now.

There isn't much written up on how to compute the higher chiral homology of vertex algebras. Besides the original work of Beilinson and Drinfeld that covers in particular the universal cases of Virasoro and Affine algebras you won't find much more. There's an unwritten theorem by Gaitsgory that proves the vanishing for simple Affine algebras at integral level.

In the case of elliptic curves and in the limit of the nodal elliptic curve, together with Van Ekeren we related the first chiral homology group to the first Hochschild homology of the Zhu algebra (just as comformal blocks are related to the zeroth Hochschild homology) in https://arxiv.org/abs/1804.00017

Unfortunately that relies on a technical condition of vertex algebras being "classically free", that is that their classical limit is isomorphic to the arc algebra of the C2 quotient. This was shown to be the case for minimal models $Vir_{p,p'}$ if and only if $p=2$. And some simple affine algebras at integral level.

We can now relax this condition to vertex algebras to the case then the classical limit is a quotient of the arc algebra by a finitely generated differential ideal. This is still very hard to prove. The only known example of this to my knowledge is the Ising model $Vir_{3,4}$ in

https://arxiv.org/abs/2005.10769

Using this we can now prove the vanishing of the first chiral homology group of an arbitrary elliptic curve (not necessarily the nodal limit) with coefficients in either the 2,p' minimal models or the Ising model. As well as recover some of Gaitsgory's results. That article should come out soon, it's being finalized for a while now.