A virus is basically not a full cell. It doesn't do much until it enters a cell where it essentially feeds information to make the cell start doing things. Like creating more of the virus. It can't do that by itself. It relies on the host cell to do all the processing.
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.