Can anyone give a clue as to how this dart game works, or just how to beat it? I keep clicking when it's my turn and my darts just keep missing the board entirely.

Ok, Fil, you can apparently bypass the dart game--I found a walkthrough that said you could play darts, "Or, if you prefer, ask Publican about his disgraceful military record, and allude to the photo of the pachyderm and the photo of his mother..."