A good start would be by installing C2 and using their platform example as a start. For the LAN part, you can purchase something from the Scirra store for multi-player.
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For individuals to see their score, you fortunately don't need a database. You can use local storage:
https://www.scirra.com/forum/how-do-i-do-a-high-score-with-local-storage_t150975
If you want the scores to be shown for different players, then you are going to need another mechanism. Easiest would be to make an AJAX call to your web server where you store and retrieve the scores:
https://www.scirra.com/manual/107/ajax