I have no idea how to code background and weather extensions, so any ideas?
What extension? You did not tell us. If we knew, someone could provide specific advice. 
First learn how to use the weather extension, make a Project to extract weather information without the BackgroundTasks extension; then learn how to use the BackgroundTasks extension. Then put what you learned together might be a reasonable strategy. Read the documentation for each extension and experiment.
Notes:
You do not have to have an extension to compile weather data from the Web.
- MIT Ai2 crashing - #8 by ABG
- Weather Information Extension
- discussions in the community about using weather information Search results for 'weather' - MIT App Inventor Community
The extensions I have issues with are the Background Tasks Extension and the Weather Extension
A simple search has not revealed any extensions called 'Weather Extension'. Who wrote it? Is it this one [Broken it no longer works. Use a weather api instead] Weather Mate ☁ [free]
Have you experimented with your weather extension without the Background Tasks Extension? If not, do so and tell us if it is working for your Android to extract the data you want. You need to learn to use the weather extension (or learn to use a weather api with Blocks ) before you have any chance to use it in the Background (which may or may not be possible). At the moment we do not know what extension you are using and what you have tried.
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It is weather mate