We don't know.
It has been deliberately kept vague, as this interview with (then) showrunner Sera Gamble shows:
Interviewer: ...Do you mean Chuck is the obvious
answer, or that you really did leave it open?
Sera: Don't ask me to squish the God debate! How many TV shows can say their
fans are talking about THAT stuff?
There are a variety of fascinating things that support the argument though:
- Chuck created the Supernatural books, which mirror the lives of Sam and Dean. When he first "discovers" that his books are real, he says:
Chuck: Well, there's only one explanation. Obviously I'm a god.
You're not a god.
Chuck: How else do you explain it? I write things
and then they come to life. Yeah, no, I'm definitely a god. A cruel,
cruel, capricious god. The things I put you through – The physical
Chuck's writing name is Carver Edlund, which is a mashup of Jeremy Carver and Ben Edlund, who are Supernatural writers - just as they are gods over Sam and Dean and the Supernatural show, Chuck could be THE god within the show.
We learn in Dark Side of the Moon, from the angel Joshua, that God is in fact on Earth.
We learn in Sympathy for the Devil that Chuck can feel the presence of the angel's in the house. However, we later discover from Kevin Tran (a true prophet) that this is not in fact one of the prophet's powers.
In Swan Song, Chuck is called by Dean, answering the phone "Mistress Magda". This is in reference to a newspaper article on his desk, with a bikini-clad woman called "Miss Magda" he had supposedly invited over. This is also a name very similar to Mary Magdalene, a followed of Jesus.
Also in Swan Song, when the day is "saved", Chuck, dressed all in white, comments on how hard it is to write a good ending, before magically disappearing.
There are also a variety of fascinating things that go against the argument:
According to Castiel, in Good God Y'All, Dean's amulet will burn hot in the presence of God - but Chuck shows no reaction to it. (Having said that, Joshua tells Sam in Dark Side of the Moon that "Magic amulet or not, you won't be able to find him".
Castiel doesn't recognise God or anything higher power-esque about him.
Other Interesting Info
The actor who played Chuck, Rob Benedict, claimed the character was God and that other members of the cast had asked him how it felt to be God. Footage of that is here:
Believe what you will. It doesn't look like we'll ever get a definitive answer, with the show creators preferring to let fans speculate.