Tom Taulli - Before your company receives an infusion of equity capital—whether from a venture capital firm or an angel investor—an investigation into your operations usually takes place. The process is commonly known as due diligence, and it can be grueling since the investor uses it to try to uncover weaknesses, potential liabilities, and risk exposures.
I propose the following: why not perform a similar investigation on your potential investor? After all, she will essentially become your long-term partner. Knowing her background, in terms of prior deals, industry expertise, and executive stints, is important.
Assuming the investor will allow you to engage in reverse due diligence (and it's a warning sign if she won't), how you start the process depends on whether your investor is a VC or angel. It's generally easier with VCs, since there will probably be more information available on the firm than on an individual or an angel group.