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.