(1) is incorrect. (2) and (3) are correct. I usually use (3).

1. How's the weather like? - Not correct 3. What's the weather like? -correct 2. "How's the weather?" -I have heard people say this, but it's not that common. It's more like "How's the weather in Spain?" or How's the weather where you live?" or "I heard there's a storm there, how's the weather?!" Siri: "What's the weather like for today?" or "What temperature is it?" or "What is the weather forecast for today?"