How do you go from virus infection to disease?
In viral infections, the disease (on the cell/tissue level) is mainly due to immune response that results in inflammation, for example, of the throat mucosal lining, which can trigger symptoms, such as dry cough.

Each symptom has its own trigger. For example, in flu:


  The mucous membranes lining the nasal and sinus cavities can become
  inflamed when infected with the flu virus. This results in increased
  pressure around the eyes and face, which may lead to a magnified
  headache (National Headache Foundation).


and


  The immune reaction to the viral infection and the interferon response
  are responsible for the viral syndrome that includes high fever,
  coryza, and body aches (StatPearls).


How viruses cause symptoms is a broad question - it depends on the virus and exact symptom, so just few more examples:


In viral pneumonia, immune response triggers inflammation in the lung alveoli, which can result in shortness of breath (StatPearls).
In AIDS, HIV virus destroys CD4 lymphycites and thus impairs cell-mediated immunity, which makes the infected person vulnerable for infections, so most symptoms are from those infections, for example, oral thrush due to overgrowth of Candida yeasts in the mouth (MerckManual).


Viruses usually do not cause disease or symptoms by merely changing the function of the cells, but by damaging them, so it's more about "injury" than just a changed function. The damage of the cells and the viruses themselves trigger inflammation and most of symptoms usually originate from it.