It doesn’t have to be anything complicated. It can be as simple as being given a task by his patron to perform before he can take another level in Warlock. These tasks can be as benign or foul as you want, but I feel like lower levels would warrant something easier to swallow, while the more powerful they get, the more difficult, or heinous, the request.
Since his patron is the Undying, it doesn’t have to be evil, per se. But if it were a fiend patron, then at lower levels it could be “kill a minion of a rival of mine” but in order to go from lvl 19 to lvl 20, it would be something like “sacrifice a baby.”