Usually, to have someone do something = to cause / force them to do it.
But in OP's context it's more like describe / depict someone as doing / being like something. This is very much an idiomatic usage that learners should be cautious about adopting (it can occur in formal as well as informal contexts, but it's never necessary, and it's easily misused).
I think the reason this usage is "easily misused" is connected to the fact that it's effectively derived from the more common cause / force sense.
Consequently, the depict sense carries overtones of the "subject" being in control (he chooses to depict the "object" in a certain way, forcing his description to apply). This can sometimes imply criticism of the subject (for choosing to apply a description that the current writer / speaker doesn't agree with).
You can see that implication in OP's actual example, where Billy clearly doesn't agree with the way he's been depicted!