I'm pretty certain this will look like a dumb question, and it probably is, but I couldn't find an answer. If a series is named after its main character, should the tag be listed as a character tag or a series tag?

It's quite simple, my friend. If a series is named after the character, such as Naruto, the character itself would be Naruto_Uzumaki, while it would it would still maintain the tag of Naruto, implying the series.

What if the last name is included in the title, as with Harry Potter?

Well, as it turns out, it would be Harry_James_Potter, as he has a middle name, and the series itself would be Harry_Potter.

Now in the case that weren't so, it would be Harry_Potter for the character and Harry_Potter_(series).

Hope this helped clear things out.