Asmodeus is a master schemer, his (its?) most cunning plans span centuries. I would play into this and realize that my puny human intellect couldn't fathom his brilliance. Let him be this haughty, but always hinting of greater insight into his devices. When his minions are encountered they seldom know the same thing, and often contradictory things. Again, his plans are intricate, elaborate and for him only.

If you want to hand out tasks let them be for no specific reason and for various agents of Asmodeus. Menial tasks for a fresh acolyte. Or perhaps hold off on tasks, "soon" agents will say Soon dedication will be proven. This is just Asmodeus letting the character think the devil is on them, to let them stew in their own fear.

Perhaps Asmodeus even want the character to break his vow and thus forfeit his soul. Another soul to power the infernal machines of war.

Or perhaps Asmodeus doesn't even care? The character is handed off to a local agent as a reward for the agent having accomplished something. Now we get into politics and very understandable motives. Here I would pick something not really opposed to the party at large, nothing really distasteful. But the agent stand to gain more than what is apparent.

More specific than that you may have to look at the party itself and what their drives are. And the local setting.

The traditional D&D devil arc revolves around a "Devils Contract". The classic D&D trope is that devils contracts are long and riddled with fine print that can come back to bite the PCs - remember, devils are evil, but they are lawful evil.

In the case of your PC, I'd probably give them some boon, but some other curse to balance it out. A very RAW solution might be forcing them to trade a level in their class for a level in "Devil's Pact" Warlock.

If you're looking for modern inspiration, check out the "Wyll" character from Baldur's Gate 3.