Part of the problem is the assumption that simple, single-cell life forms are simple. In actual fact, they are not so simple. They have moving parts, they have a control system that directs the operation of those parts, and a basic logic blueprint that directs how they are formed, how they operate, and how they fit themselves into their environment.
What distinguishes "life" from "non-life" seems to be a philosophical problem when we don't have a powerful enough microscope, and we can't see the moving parts. But our very basic lifeform has a built in logic that guides how it does what it does. When the signals that trigger its actions are no longer interpreted correctly, it starts to fail. When the signals are no longer generated at all, we would say that it is dead.
The signals, be they chemical or electrochemical in nature, and its response to those signals are the indicator that it is alive. As others have pointed out, the DNA (which is presumably where the logical blueprint exists) is still there even after it is dead, but the signals that keep the little mechanism working have stopped.
I realize that I am treading in way over my head. Cars are mysterious to me, but they are not magic. They have moving parts and a logic that controls their sequence of operation. There is a continuum I suppose between an engine and a raw block of iron ore, but the relationship between the two should not obfuscate the fact that an engine is not ore. But cut the cord from the ignition coil and it stops dead in its tracks.