Suppose I cast cloud of daggers on a creature with 0 hit points that already has 2 death save successes and 2 failures. Then the creature starts its turn.

Does the creature take the damage from the cloud of daggers first, resulting in their third failed death save and killing them? Or do they get to roll the death save and potentially be stabilized (and maybe even recover 1 hit point) and then take the damage and now have 0 successes and 1 failure (0 if they had recovered a hit point)? Do both the damage and the death save happen at the same time, and if so, what happens?

If you go by the ruling offered in Xanathar's Guide to Everything the creature whose turn it is can determine which effect happens first when they share the same trigger like "start of turn".  In your case the creature would probably choose to roll the death save first.

A DM may rule different in favour of the player: in your case, since you are a player casting the spell against an NPC, then the damage occurs first. This will give the player more satisfaction and make the combat faster. 

Ultimately: determined by DM but there is a precedent as described above.

Thank you. My DM would probably go with the precedent described as there are far better ways to finish off a creature in this situation than cloud of daggers. Interesting to think about, though.