As someone who has done a significant amount of professional magazine writing, I would say that this depends entirely on the audience from whom you are interviewing her. If you were doing an interview for a serious news outlet, you would ask different questions than you would for a fashion or celebrity lifestyle magazine. 

The first issue you might want to discuss with her for a somewhat serious article would be the difficulty of holding power as a woman in a strongly patriarchal society. You might ask whether her relationships with the various powerful men in her life were politically necessary or whether they were motivated by actual love for the men.

Next, you might ask about the tensions between Greeks (including her own Ptolemaic dynasty), Romans, and Egyptians in Egypt. one interesting question is the degree to which she considered herself a Hellenic or Egyptian ruler and whether in private she tended to think in Greek or Egyptian.

Another serious political question would be how Egypt could sustain its unique cultural identity and traditions under Roman rule. Did she feel her relationships with Roman men were a betrayal of that identity or necessary to preserve it?

Wow. That's quite a project. Cleopatra led such a full life that there are many questions I would like to ask. For instance,
1. What was it like being one of the only women to be a head of state during her life?
2. What did she think of Julius Caesar?
3. Who did she love the most, Caesar or Mark Anthony?
4. Why did she feel she had to kill herself?
5. She was noted as a great beauty during her time. Did she any any special beauty secrets?
6. How many languages did she speak?
7. How was she educated?
8. What was her biggest goal in life?
Those questions should get you started. You might want to check out the links listed below.