Does an ability that says 'whenever a creature dies' trigger from a creature dying during the cleanup step?

Normally, triggered abilities that say 'whenever a creature dies' will trigger whenever a creature is put into a graveyard from the battlefield, regardless of when during the turn that happens — including the cleanup step. This is because 'dies' simply means moving from the battlefield to the graveyard (CR 700.4), and the trigger condition doesn't restrict itself to specific steps or phases.

The cleanup step is special: players normally cannot receive priority during it (CR 514.1, 514.3). However, if a triggered ability triggers during the cleanup step, the game must give players a chance to respond. CR 514.3 states that if any triggered abilities trigger during the cleanup step, those triggers are put onto the stack and players receive priority — a new cleanup step is then performed afterward.

State-based actions are checked at the end of the cleanup step if damage was dealt (CR 514.2). If a creature happens to have lethal damage marked on it at that point (e.g., from a Wrath effect resolving earlier, or from a 'until end of turn' effect expiring), it dies, and any 'whenever a creature dies' trigger fires. An extra cleanup step then occurs to handle that trigger.

Concrete example: Your opponent controls a 1/1 creature enchanted with an Aura granting +2/+2 until end of turn. At the start of the cleanup step, the +2/+2 bonus expires (CR 514.2), making it a 1/1 again. It had 2 damage marked on it, so state-based actions kill it. Your 'whenever a creature dies, draw a card' ability triggers, an additional cleanup step is added, and you draw a card.

So yes, 'whenever a creature dies' triggers do function during the cleanup step; the game simply generates an additional cleanup step after those triggers resolve (CR 514.3).