Let me add that the second can also be considered an adjective. Let me explain with a simpler example. Consider the sentence "this engine runs hot" Heat is a property of engines, not running. It is the engine that is hot, not its running. The sentence has the same meaning as "this engine is hot when it runs". If you used the adverb "hotly" the sentence would sound wierd: "this engine runs hotly". Next, consider our example: "he runs fastest in the class" This one confuses us more because fastness can be a property of a person or of running. "Fast" is one of those words that can be used as an adjective or an adverb. If you think of it as an adjective, then you are describing the boy, not the running. He is a fast boy. He is the fastest boy. The sentence is then the same as "He runs as the fastest (boy) in the class". But you can also think of it as an adverb.

Great question. In the second, it's an adverb. As for the first, it seems to me that "fastest" is not an adverb there, and it does not modify "runs". Instead, I think "fastest" is an adjective and fits into "the fastest [one] in the class" which acts like a noun, as in the sentence "He is the fastest in the class". I would interpret the first by inserting two missing but understood words: He runs [as] the fastest [one] in the class. The article "The" requires a noun, does it not? If you stick "the" in front of an adjective, like "the largest" or "the loudest" or "the fastest" something always seems to be missing: "the largest WHAT", "the loudest WHAT", "the fastest WHAT". If the WHAT is not stated, it is implied.

It's an adverb in both of your examples. If you want to make it an adjective: Of everyone in the class, he is the fastest runner.