Let's go through a number of the criteria that are used to decide if something is alive:
1. Does it grow? Yes, the universe is expanding.
2. Does it reproduce? No, the universe does not reproduce itself.
3. Does it maintain internal state (homeostasis)? No, the universe is cooling as it expands as would be expected from thermodynamics.
4. Does it move? It isn't even clear what it would mean for the universe as a whole to move.
5. Does it have a metabolism (ingesting things for nourishment and using the components to build itself)? No, the universe is not ingesting anything from outside of it as far as we can tell.
6. Does it respond to stimuli? Again, no, there are no external stimuli for the universe to respond to.
So, for this list, there is one answer consistent with being alive and the rest are not.
Conclusion: the universe is not alive.