Learn French: Ce qui me dérange, c'est chaque blague vulgaire.

We use Ce qui because "what" is the subject of the verb dérange (it is doing the bothering). If "what" were the object of the action, like in "What I hate" (where "I" is the subject), you would use Ce que instead.

Why do we need the "c'" in "c'est"? Doesn't that mean "What bothers me, IT IS every vulgar joke"?

Yes, it translates literally to "it is". In French, when you use a Ce qui or Ce que phrase to emphasize something at the start of a sentence, you must introduce the second half with c'est (or ce sont). You cannot just use the verb est on its own here.

Why doesn't the word "chaque" change to match the feminine noun "blague"?

The word chaque (meaning "each" or "every") is invariable, which means it never changes its spelling. It always ends in an "e" and is always used with a singular noun, regardless of whether that noun is masculine or feminine.

Why is "me" placed before the verb "dérange"?

In French, object pronouns like me must go directly in front of the conjugated verb they belong to. So the English "bothers me" flips to become me dérange.