I am a little confused honestly about this whole thing. What do you think is the fastest time, but I doubt this is correct. Really any kind of break down of sentence structure or meaning would be very helpful. Thank you!
Is this sentence easier to understand now?
时间过得最快 => Time passes the fastest. This is relatively easy to parse, I reckon.
The first part of the sentence is trickier. "X的时候" means "when X". At first glance X seems like it might be the whole 你认为做什么 clause or only 做什么. But 认为 is used to talk about one's opinion, so 你认为做什么 as a standalone clause doesn't make sense. That means "做什么事的时候" ("when doing what thing") is correct; and the question is "What is it that when you do it time flies the fastest?"
An answer to this question would be 我认为玩电脑的时候时间过得最快 (I think time flies the fastest when (I'm) playing computer games).