1) The infected cell, is no longer the same as the organism's other cells. If it's a single celled organism, the infected cell no longer the original organism.
2) Once the "infected cell" starts shedding the virus, that's when the virus is alive so to speak. the actual virus microbe thats shedded can be thought as a sex cell of sorts. (Like pollen or mammalian sperm or eggs, etc.)
Disclaimer. I'm not a virologist and as far as I know, *most* virologist conclude (last time i checked) that viruses are non-living microbes
If you think of the cell as a computer and the virus of some kind of information carrier (tape, cdrom, etc.), you can answer the question by asking yourself whether a cd-rom is capable of doing computation. Which would be a good analogy for being alive. To which of course the answer is no.
So, by that analogy, viruses are not alive. They don't act by themselves. In the same way DNA is not alive. It's just a passive protein.