Well, you're correct not to split the People table.
the first thing to do is add a Roles table, that will contain role id and role title (each column should be unique - you don't want 2 different ids for the Actor role...)
TblRoles
RoleId RoleTitle
-------------------
1 Director
2 Writer
3 Actor
Then you add a PersonToSeries table, that will hold it's own id, the person's id and the series's id. This table will hold every person ever working on that series, being a regular staff member or a 1 episode guest.
TblPersonToSeries
PTSId PersonId SeriesId
---------------------------
1 1 1
2 3 8
3 4 7
The next table you will need is a PersonToEpisode table, that will hold the PersonToSeries id and the episode id, and the role id. many popular movie apps like Cinema APK are using it
In this table you only need to keep integer ids so it's very light weight, and you will specify for each record in PersonToSeries the episodes it is relevant for.