"ce que" (or "ce qu'") is one of the four indefinite relative pronouns. It means "what" or "which". It is used as a direct object (in this case, of the verb "savoir" -> "sais", meaning "know").
For grammar nerds, there are three more indefinite relative pronouns, all of which mean "what" or "which": "ce qui" (subject), "ce dont" (object of "de") and "quoi" (object of other preposition).
I wonder if one of our Francophone colleagues can comment on whether "...what has become of him" is within the scope of this sentence.
In English, "...what he has become" addresses some personal transformation of the person under discussion. He has become a judge, a criminal, a grandfather, whatever. "...what has become of him" is more general; in this case you'd be saying that you have no idea where he is or what he might be doing.
Not exactly, it means you don't know where he is as well as what he has become, but it might be correct anyway. http://context.reverso.net/translation/english-french/I+don%27t+know+what+became+of+him
Can someone shed light on why, for the English translation, the response uses the perfect past on this sentence (has become), whereas many of the other sample sentences are only translated using simple past (became)? Does the perfect tense not "exist" in French as it does in English, or is there a nuance to this sentence in French that limits the translation to best being represented with perfect past in English?
In English, the versatile simple past tense can be used to replace any past tense, including the present perfect tense which is actually a past tense that has recently happened or that affects the present. In French, the passé composé is not limited to the English present perfect and often uses a form of “avoir” to form it, but other verbs use “être” including “devenir”. The forms “has become” or “became” or “did become” could all be used for this sentence, but I don’t think that I would use the emphatic form for this particular sentence. It is helpful when the perfect tense is used, so that you can become familiar with the fact that in English we use a form of “to have” to form this tense, but in French “to be” is used with this particular verb.