I haven't used Tarot cards to gain insights or guidance related to the weather, but I can see how this could be done. Consider looking at the card's imagery and seeing what it could signify - if there are images of water, for example, that may be an indication of rain. Then look at each card in relation to the others, considering how the four elements, Earth, Air, Fire and Water, might be represented.
One idea for a spread might be to lay out a few cards for each season - one card for spring, one card for summer, one card for autumn and one for winter. From there, you can meditate on each card and what it means to you in terms of the current season. You might also look at the suit of the card, cups may represent rain and storms, wands may signify strong windy days, and swords can point to cold or icy weather.
I would also suggest focusing on the major arcana cards as well, as these cards represent larger cycles and tendencies. If you need even more guidance, you might incorporate a question such as, 'what will the next month's weather be like?', and focus on the cards in reply to that question.
Overall, I think the best way to incorporate weather into Tarot readings is to let your intuition guide you. Trust in yourself, and take the time to meditate on the imagery of each card. This can bring deeper insight and guidance than just making literal interpretations of what the cards represent.