Legality of doing private or scholastic virology with human pathogens?
tl;dr I think that doing this unofficially, at home, might be no big deal, but doing it in an institutionally supported lab (or any kind of institutional setting, e.g. if you wanted to publish a paper from the results) could involve lots of hassle and red tape.

I'm not a microbiologist, but ... technically I think this would fall under the category of BSL-2 - via Wikipedia:


  Biosafety level 2 is suitable for work involving agents of moderate potential hazard to personnel and the environment.[9] This includes various microbes that cause mild disease to humans, or are difficult to contract via aerosol in a lab setting.[11] Examples include Hepatitis A, B, and C viruses, human immunodeficiency virus (HIV), pathogenic Escherichia coli, Staphylococcus aureus, Salmonella, Plasmodium falciparum, and Toxoplasma gondii.


BSL-2 requires (among others) access control for labs; special safety training for all lab personnel; use of containment equipment for procedures that may cause infectious aerosols or splashes ...

It may seem a little crazy to lump a common-cold-causing rhinovirus with HIV, but BSL-1 (the mildest level of infectious agent control) is restricted to  


  well-characterized agents which do not cause disease in healthy humans


(emphasis added). Note that even BSL-1 requires lab safety protocols beyond those for non-infectious agents.

A bit more internet searching on BSL classification for rhinoviruses (the typical cause of the common cold) suggests some variation: e.g. 


Google search for "rhinovirus BSL" brings up a bunch of hits from the American Type Culture Collection (ATCC) for a variety of rhinovirus isolates that range from BSL-1 to BSL-2
the Public Health Agency of Canada classes rhinoviruses as BSL-2