I would love to hear thoughts and ideas of demonic and nondemonic activity that members of an Asmodeus cult could be up to, as I am currently plotting one of the bigger long-term mysteries in the city where my players are based. 
I plan to have several demonic-themed encounters, making the players think... okay there is a LOT of demonic activity in this city, why might that be? Gathering leads and eventually figuring out the existence of the Asmodeus Cult. 

Until now, the players have come across some imps and a devil trying to kidnap the children of a noble home, the issue was resolved, and the characters left the house without suspecting or investigating anything further. 

So the cult is meant to be mainly rich and powerful people worshiping Asmodeus as a means to stay rich and powerful. Not opposed to using demonic power to tip the scales to their advantage. All the while they have a secret society doing underhanded dealings amongst each other, nurturing a healthy survival-of-the-fittest-mindset, and stepping on those foolish enough to trust them. 
So... Idears for what kind of shenanigans they could be up to on a day-to-day basis, just to jumpstart my imagination, is very much appreciated :) 

Asmodeus is all about luring the souls of the wealthy and powerful so I think a lot of the cults time would be dedicated to spreading his influence. 

Hosting 'Eyes Wide Shut' style balls and parties to seduce people with political clout by showing them all that worshipping Asmodeus has to offer could provide a cool infiltration scenario.

Training and sending people to take up positions of power - this could be as mundane as running (seemingly benign) schools teaching politics and statecraft, perhaps even an Oxford/Harvard style university for the rich.  Students discovering (and disagreeing with) the ruse would disappear - could make for an interesting mystery scenario.

He also gains power from the faithless, so setting up the worship of fake deities through seemingly charitable deeds could be a nice misdirection.

More overtly, having some spined devils (or other) attack the cultists' rivals (Asmodeus keeping his promises) could certainly help provide some obvious clues to track down.  Again, this could be fairly mundane - attacking a rival trader's caravans, destroying warehouse stock or scaring off the stevedores and dockhands so the rival cannot unload their ships.  Even just sending imps or other minor fiends to disrupt a rival's sleep and routine during an important time could provide what at first seems to be a regular side-quest but connects to the main story.

And again, thinking of the faithless, you could have some poor paladin being hounded (family dying, home in flames, dishonored in some way) in an effort to have him forsake his oath.

Anyway, those are some ideas I had - hope they help with the jumpstart.

Never forget that by far the most important thing to a game of Dungeons and Dragons is a dungeon and a dragon.  So start thinking about what dungeon these cultists are preparing for the players, that’s the meat-and-potatoes of your game right there.  

Think about what the cult’s dungeon would be, an abandoned castle with an underground complex comes to mind.  From there you can start making the floors and then furnish and populate the rooms with encounters, and be well on your way! 

h'o'w's'a'b'o'u't(i wasnt sure where the apostraphe went) the cult randomly picking a family(or person) they know, and becoming "friendlier" with them. inviting them to golf, the country club, offering to pay for expenses(that other members of the cult caused of course), until the person(or family) become dependant on the cult, and then pressured into joining the asmodeus cult and repeating the cycle. MUAHAHAHAHAHAHAHAHAHAHAHAHAHA