What is the proper usage of the phrase "due diligence"?

A lawyer referring to the process of investigating a potential merger/investment might say:

We need to perform due diligence.

There is also business buzzword of "due diligence", derived from the legal meaning to mean the level of care/attention that one would reasonably be expected to take in this situation. In my (American) experience, this is commonly used in the business world as an idiom:

We need to do our due diligence.

One can also consider the literal meaning of the words. "Due + Diligence" = "Appropriate Attention", so one could say:

We need to give this investigation its due diligence.

Looking at the OP's original three phrases according to this guidance:

We need to do our due diligence to investigate this. (correct, business usage)

We need to use due diligence to investigate this. (incorrect)

We need to perform due diligence to investigate this. (most likely incorrect, unless you're dealing with a business merger)

"Due diligence" is a legal term to describe when one has exercised an appropriate level of caution or investigation prior to acting or making a decision. To "do due diligence" is an attempt to use the legal term in a grammatically inappropriate way.

A more appropriate ways to express oneself, if one really must force the legal term into one's sentence, would be to "exercise due diligence." Personally, I would prefer if people could merely express themselves intelligently while conveying the same message without forcing the legal term such as to "be duly diligent."