Cells read DNA in the nucleus to 'know' what to do. The DNA clones itself and then the DNA is put into the newly produced cell, this means a new cell has been produced - this process repeats. The virus will take advantage of this, and latch onto a cell, and penetrate the cell membrane with it's **. The virus will then continue to inject extra DNA into the cell, through the membrane. This will cause the cell to create clones of the virus, instead of clones of the itself; however this means that the cell cannot reproduce itself, and therefore, after filling up with viruses, will 'pop' and release many duplicates of the starting virus, back into the body, to infect more cells in the future.
Credit to "http://micro.magnet.fsu.edu/cells/virus.html"
"Pandemic: A Horizons Guide" is a long video, explaining the pandemics that have occoured in the past.