Even for someone who can't see it, it can be definitively informative that there is an image that contains the person in question. It is important to keep the text relevant to the context and to why the image is presented there. If the person is fishing, but the image was chosen only because it is the person in question, just give the name and don't mention that he is fishing. If the image only represents that it is about a person and shows an illustration of a person but not the person in question...
...this illustration should have an empty alt attribute or is a valid case for a background image. This article contains a very good explanation of when and how to use the alt attribute with very good examples.
In your case it's a picture of the person mentioned and should have an alt text. But it should be an image element to be able to provide this information in the first place. The element has the semantic meaning that an image is presented, why it is presented can be noted in the alt attribute but also in the surrounding text. If the element is closed in by another element that contains no describing text, the alt attribute should have this text. If the element is in the same element as the describing text, that text is sufficient and the alt attribute should be empty (but not omitted!).
Both are correct:
<a href="jdoe.png"><img src="jdoe_small.png" alt="John Doe"></a>
<img src="jdoe.png" alt="">
In your case the background image will not be noticed by people using screen readers. So you are missing the semantic information that an image is present. Solving this means you have to add screen reader only text telling that there is an image and that it contains the person in question. Or you can omit that information, but besides that this is discriminative, it is also completely unnecessary. Why not just replace the background image with the semantic image element? Any other solution is harder to implement and is possibly confusing for the user.