This is a challenging question, but I think the best way to answer it is by examining what virus can and cannot do. By doing this, we will be able to argue that they are both alive and not alive. We can argue that virus are alive, because they do exist in every facet of life. More importantly, viruses are able to mutate, adapt, and replicate. This adaptability is a strong requirement of life. Without this ability, no organism, including humans would be able to exist for very long. Moreover, if we follow something as simple as the flu, for example, we will see that new viruses are found all the time.
What virus cannot do is exist by themselves. For this reason we can argue that they are not alive. Without a host or a cell of an organism to live in, virus could not exists as far as medicine and science is able to tell. From this perspective, we can say that they are parasitic. Left alone, they simply cannot reproduce.