Am I able to deduct the cost of Internet bills if I attended online classes?

YES.

If you only purchased internet to fulfill the requirements of online classes, yes. 

The more realistic amount would be the expense for the portion you use it for school, so if 80% used for school, claim 80% of the cost for the time/months you attended the classes. 

No, if you are only using the internet you already had before enrolling in classes.  If you purchased an upgrade, only the additional cost would be a qualified expense. 

 To be eligible for the tuition credits, the course must be taken at "an eligible institution". The school should be able to tell you if it is an eligible educational institution. In general, an eligible educational institution is an accredited college, university, vocational school, or other postsecondary educational institution, including accredited, public, nonprofit, and proprietary (privately-owned, profit-making) postsecondary institutions. Additionally, in order to be an eligible educational institution, the school must be eligible to participate in a student aid program administered by the Department of Education. If they issue a 1098-T they are probably an eligible institution.

Enter your school at the link below, to see if it's on the dept. of education list.

   https://www.savingforcollege.com/eligible-institutions    https://ope.ed.gov/dapip/#/home(download)

Schools that only offer online courses usually do not qualify.  Online classes taken at schools that also have  classroom instruction usually do count.