Leaflet appears to be focused on GeoJSON and tiles. You would need to put your raster data in GeoJSON format along with lon lat information. Depending on the size of your raster data pixel count, one large GeoJSON file with all the weather data could swamp a web browser's memory. You have to have some way of working with a bounding box of the data either in your Leaflet script or from a server service. You want to look for marker information in Leaflet. Also note that there is a GeoCVS Leaflet plugin. There's some links to get you started.
How to interact with leaflet marker layer from outside the map
Leaflet - How to find existing markers, and delete markers?
How to plot tweets by location in real time on to a map e.g. leaflet