"After his performance, critics knew (who,whom) the next musical star would be."

In order to decide whether to use "who" or "whom" you need to parse the sentence to determine what part of speech it is. First, eliminate the initial prepositional phrase "After his performance" because it does not affect the rest of the sentence. This leaves you with:

"critics knew (who,whom) the next musical star would be"

Words such as know, think, say all follow the rules for reported speech, and thus you should reshape the phrase into direct speech to determine the function of the pronoun:

"critics knew: the next musical star would be (who,whom)"

The main verb of the reported sentence is "would be", a form of "to be" and therefore takes a complement in the nominative case rather than a direct object. Therefore the answer is "who."