This is the character I did last week.
Would it be correct for an artist to use "did" in the sentence above?
She is talking about the (game) character as an assignment she finished last week.
Personally, I'd use the verb "to create"...
Do can stand in for an awful lot of other verbs. You might replace "character" in the quote with web page for a web designer, or portrait or landscape for a painter, or article for a journalist. What do stands in for depends on context. Sometimes it's there for brevity or to avoid having to think about what word to use, and sometimes it's there to avoid constructing a much more complex construction where what they did wouldn't be expressible as a single verb.
That last category applies here, I think. Without knowing the specifics, she may have conceptualised the character, done sketches, concept art, modelling, posing, animating, and if she's doing more than just the art on it, character profiling, back story, and so on. All of that can be wrapped up nicely with the verb to do.
Don't use it about actual people without extreme caution, as absent other context to make clear what you mean (such as someone whose job involves investigating/examining and writing reports on people), "to do a person" often refers to sex. Obviously that would not be the case in your example, however.