Is "ciel" like the English word "heaven", which does not need an article, or is it like "sky" where an article must be used? Does it depend on the context?
In French, the English "no article" rule doesn't exist.
Instead, we divide words into 3 categories:
When you are talking about a definite noun, you always need to put a definite article, no matter the word type:
"Le chat est mignon."
"Le riz que j'ai mangé était délicieux."
"Le ciel est bleu aujourd'hui."
"Ton grand-père a rejoint le Ciel à présent."
The only distinction between countable and not countable nouns is made with indefinite and partitive articles:
"Un chat est devant la maison."
"Je mange du riz."
Abstract nouns such as "ciel" are a bit more tricky in this particular case (there is an interesting discussion about it here in French). However, in my opinion, the only situations where you would put a partitive article in front of an abstract noun are the situations where you can consider this noun as a non-countable noun.
The comparison with the English grammar won't help you here, as the articles work differently in the two languages (see Driblou's answer for more details on this point)
In French you would usually see "le ciel", when talking about the sky, or eventually "le Ciel" with a capital when talking about heaven.
You can have the indefinite article "un": "un ciel" when you are talking about the specific aspect of the sky at one moment or one place. It works the same as in English:
Ce matin, sous un ciel bleu, je marchais vers la mer. / This morning, under a blue sky, I was walking toward the sea
Finally, there is the old-fashioned interjection: "Ciel!" Typically:
Ciel, mon mari !
is the iconic exclamation of a woman finding his husband cheating on her in vaudevilles.
As it is an interjection it doesn't take any article.