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Let $A$ with $d$ be a differential graded associative algebra ($DG$ algebra).

What is the definition of the Hochschild complex $C^*(A,A)$?

There should be a Hoshschild differential and a cup product, but I can't find the reference that specifies the formula. Does any one know?

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A formula appears here, probably a repeat question... – Daniel Pomerleano Aug 4 '12 at 18:36
Or try this one:…, page 4, since you also wanted a formula for cup product. Sign conventions often differ and it's important to have ones which are consistent. – Daniel Pomerleano Aug 4 '12 at 18:46… mentions them – CSA Apr 6 '14 at 18:53
@DanielPomerleano Would you be able to fix the dead link in the second comment? Thanks. – Todd Trimble Apr 20 '15 at 18:44

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