Is the following the correct grammatical way to use 'persona'?
I wonder what persona you have of me?
It's grammatically correct—verbs, nouns and pronouns correctly deployed—but meaningless, as per "colourless green ideas sleep furiously".
About the only interesting thing happening grammatically is that the question mark turns a plain statement about what you are wondering about into an implied question, looking for an answer to that wondering (in spoken English it may be indicated with a rise to the end of the sentence).
You do not have personas of other people, you present personas to other people. If a boy acted as a diligent student in front of his teachers, as a dutiful, considerate and hard-working son in front of his parents, as mysteriously aloof and slightly rebellious though mostly successful young man in front of potential dates, and as a hard-drinking hooligan in front of his peers, then he would be presenting four different personas.
However, while it's possible that each persona was more or less perceived as desired, it's also possible that his teachers think him an ill-behaved dunce, his parents a hopeless layabout, his peers a cowardly light-weight and those potential dates as a clumsy buffoon.
While he projects personas, they perceive impressions.
I wonder what impression you have of me?