I believe national-wide weather forecasting by computers was been done since the 1960s. As today's personal workstations are much more powerful than a supercomputer built in the 1990s, and the old models and numerical methods (not the state-of-art ones) are well-known (and have multiple open-source implementations), the calculation should be possible today on an ordinary workstation for educational purposes.
But exactly what kind of data do I need to run my own weather forecast on my computer? Is it possible to calculate my personal 12-hour weather report based on public data published by NOAA, etc? Is there a tutorial for setting up a weather forecast?