Historical weather data with machine learning? My company gave me a task to build some weather forecasting. I have now historical weather data for 10 years (temperature, precipitation in mm, humidity and etc. more than 30 features total). We need to know what probability for rainfall in South-Asian countries in particular time period (week, month)? 

Now I'm able to calculate approx probability based only on monthly averages (like if for 10 years 1st of Dec was rainy 8 times, so the probability is 80%). Of course, it's not a scientific approach at all, but sometimes we need to know rainfall probability for 3 or more months period in advance.

I was thinking to try building a neural network model, but don't understand how it might work in production? What data to give to this model after training it? Does anybody have any ideas?
For time-series modelling, one would typically look to LSTMs (or their more recent improvements). Such models are known for their long-time-memory, which is necessary here because of long term seasonality and similar effects.

As for production, there are several methods to serve neural nets, for example Tensorflow Serve.

Specifically about training once in production - you could choose to simply not do it. If your dataset has 10 years of data, and your model will be in production for more than a year, you could think about retraining it (with all the newly collected data + old) periodically, every year or so. It could be completely automated too. 

There are other methods (look for "online training and inference" approaches, which are traditionally in the RL domain). But based upon your description, the extra time you would spend to build such capability doesn't seem to be worth the gain.