How would be the weather like tomorrow?
It can be very easily considered right in everyday English or informal style. Since would is always the past tense of the modal verb will —which is only used for future— so logically its past tense, would, should be used for the past time, not the future (tomorrow, in here) grammatically.
Hence, I think this sentence has grammar issue(s). And I would say: "How will be the weather like tomorrow?"