Should I cite the interviewer or the interviewee when citing a news article?
A standard MLA method for citing a published interview is described here:


  [Interviewer]. Interview with [Interviewee]. [venue] (year): [pages]. [medium].


e.g., 


  Lois Lane.  Interview with Superman. Daily Planet (1963): 12-13. Print.


As noted in the comments, you would then refer to it in text using per the style with whatever prose is most clear and lucid, e.g., "In an interview with Lois Lane (Lane, 1963), Superman claimed that his only weakness was kryptonite."

Citing in the APA style appears to be similar, and all other styles should have an equivalent as well.