Learn French: Ce que je déteste, c'est le brushing, la laque et le volume ! - What I hate is the blow-dry, the hairspray, and the volume!

Yes! Just like the prepositions à, de, and en, definite and indefinite articles must be repeated before every noun in a series. While English often drops the word "the" in a list (saying "the blow-dry, hairspray, and volume"), French strictly requires le brushing, la laque et le volume.

Why do we need c'est in the middle of the sentence? Can't I just say Ce que je déteste est...?

In French, when you start a sentence with a phrase like Ce que or Ce qui to emphasize a point ("What I..."), you generally must introduce the second half of the sentence with c'est or ce sont. Saying Ce que je déteste est... without the c' pronoun sounds quite unnatural to a French speaker.

Why do we use Ce que instead of Ce qui at the beginning of the sentence?

You use ce que here because it acts as the direct object of the verb détester (you hate something). You would use ce qui when "what" is the subject actually performing the action, such as in Ce qui me dérange... (What bothers me...).

Is le brushing just the French word for brushing your hair?