Having a "good feeling" about a doula is the first step! I am a doula myself and have had some amazing experiences with clients and I felt like both of us just knew when we met each other that it would be a good fit. Other than that, definitely get a few references and actually check them. Also ask if she is certified, where she trained and how long she's been practicing as a doula.
You should also go over your ideal birth with her and how she would support you and your partner in that. Ask how your doula deals with complications or use of interventions in labor. Even if you are planning a natural labor, it's good to have an open mind and have an experienced doula to explain all of your options as far as interventions or pain medication (although I believe that being prepared for natural labor and having that be your intention wholeheartedly is more than half the battle!)
Ask what she does in case of a planned c-section (just in case you end up being breech later on or have another complication- how does she support couples in these situations?) You can also ask if she is familiar with the staff at your hospital/birth center/or if she has worked with your homebirth midwife. You could also ask what is included in their "package" (standard is at least 2-3 prenatal visits and at least 1-2 postnatal visits.) Ask what each visit includes (help with the birth plan, comfort techniques, etc. at the prenatals and help with breast feeding, cloth diapering, etc. postpartum.)
On the DONA web site they have a list of questions to ask potential doulas but I think that a good connection is the biggest indicator of a good match! Good luck!!