Is there a "Campaign Mode" set of rules available for Arkham Horror?
This may not be exactly what you're looking for, but check out the Scenarios FFG created for League play (here's a link to information about them on the Arkham Horror Wiki).  They do ramp up in difficulty, and the Spanish league scenarios have a continuing story.  

I'm sure you could easily adapt things so that your characters grow and develop through the games, perhaps by adjusting some of the stats or giving bonuses based on how the investigators "performed" in the previous scenario?