The definition of life is the main issue here. That is one of the great philosophical questions in Biology (and not only Biology) and is way beyond the simple Q&A format of this site. However, most definitions of life can be boiled down to1:
A self replicating entity that interacts2 with its environment.
Or, even more simply :
Life is self-reproduction with variations
In both cases, the answer to your question depends on which level you are focusing on, your resolution if you like. If you take a human cell and apply either of the definitions above, it is alive. A cell is capable of autonomous reproduction, and metabolizes substances harvested from its environment. The same holds true of you look at the human body. However, the definition will not hold for a part of the human body, for example an arm of kidney.
Consider also that a human cell is capable of making more copies of itself, more individual cells. A human body is also capable of making more copies of itself, more human bodies. They can, therefore, both be taken as alive.
There are even more complex examples like Dictyostelium discoideum an amoeba that can exist (and be alive) either as a single cell or as a colony sporting different cell types and rudimentary organization. However, the organism is alive in both of these stages. In one case, you would consider the cell to be the fundamental unit of D. discoideum life, and in the other you could consider the colony.
So, to answer your question both a human cell (as long as it is alive, actively metabolising and capable of reproducing) and the entire human body can be taken as alive. That my cells are alive does not lessen the fact that I am too. It is just a matter of perspective.
- Trifonov EN., Vocabulary of definitions of life suggests a definition. J Biomol Struct Dyn. 2011 Oct;29(2):259-66.
1 Yes, I know this is a simplification, I just don't want to be drawn in to a long winded discussion on what the definition of life is. My answer is applicable to any of them really.
2 Metabolism can be seen as a form of interaction.