Here's a very explicit answer. You can use a bar complex equally well for `A_\infty` categories and `A_\infty` algebras.

Thus consider $\bigoplus_{k=2}^{\infty} A^{\otimes k}$, put a grading on this as a sum of the number of tensor factors and of the internal gradings (maybe with a shift so we start at 0), and make this into a complex by defining `d` to be the sum (with appropriate signs) of all ways to apply an `m_j` (from the original `A_\infty` category) to consecutive tensor factors.

Now take coinvariants and homology.