How much material should be removed for a finish pass?

A finish pass is typically a shallower cut than a roughing pass. You should cut through enough material that you are meeting the effective chip thickness (you are able to create an actual chip- trying to take too small a chip will simply cause the tool to deflect and rub), but not so much material that you cannot control the machining process to the accuracy desired. This depends on metal type and end mill size. For a metal like aluminum, 0.010” is an apt dimension for a finishing pass.