Is the hacking group Anonymous a criminal group or a vigilante group for the greater good of the Internet?

What Andy said is true. Since anonymous doesn't have a central organisation its hard to determine if the group itself does "good" things. Actions they decide on usually start with someone having an Idea. They then post about it and if it draws enough (positive) attention people who are interested decide on a cause of action. So to determine if the hacking part of the group is good, since that part was stated in your question, I think one should look at the largest activities/actions that 'anonymous' undertake since those represent the point of view of the largest part of the group. Usually these extremely large events revolve around an idea that is 'good'. But I lack the knowledge about all their large actions to judge fairly.

None of the above. I actually tried to cover what they were doing in 2012, and soon found every blog post I got was flooded with 'that is not really Anonymous' or he is not really a member. It became obvious to me its more of a fashion than a movement, like asking what people who were beards want or what people who get tattoos want. There is no Anonymous, not in any structured meaningful social way.  The 'participants' are any person anywhere in the world who, at the least, thinks the mask is cool. On November 5 I went to the big march in London.  My wife was photographing it.  I brought a mask because I had just gotten one at Gezi Park in Istanbul and I thought that was cool.  I carried a large flag made from a photo my wife took so she could find me in the crowd. Several times news agencies came up and asked we what Anon was about and what were my reasons for joining and what we hoped to get out.  I was honest and told everyone I just had the flag so my wife who was a journalist could find me.  But I could have just told them anything, and since I am something of a good talker I probably could have gotten on the news.  In fact I probably could have come up with something so much better than most of the drunk malcontents who were there to face off the police I could have gotten in the papers and maybe even became a voice quoted by researchers and public figures as to Anon goals and missions. That is all it is, all it ever was.  Anonymous is just a guy or girl who likes the Internet saying the are Anonymous.