More than anything else, Napoleon functions as a plot device. The intrigue surrounding his return from Elba provides the perfect pretext for Edmond's imprisonment. His brief rise to power – the "Hundred Days" – also serves to emphasize the split in Edmond's life: as a result of his imprisonment, he misses the entirety of Napoleon's return. He's totally disconnected from life and from history.
It should also be noted that Napoleon's meteoric rise(s) and subsequent fall(s) also reinforce one of the book's most important themes: fortune. Napoleon is an incredibly powerful man – he's able to conquer great swathes of Europe not once, but twice – but he can never hold on to his power. Does he fall because of fate and fortune or because of his own actions? You can argue about it all you want, but you can't deny that it recalls the same sort of turmoil that marks the live of Edmond/Monte Cristo and the men on whom he seeks revenge.