I agree with shardian's approach. The obvious solution is to confer with him about his shortcomings.
It looks like everyone agrees that's the best course of action. Except that's probably what you were thinking of doing in the first place, I'd bet. So long as you do it the right way. And the right way is so damn hard to nail down. shardian gave one of the better ways of doing it. We had to read a book for our company, and the one thing I took from it is when you're trying to converse with someone about something that can evoke negative emotions, you don't use the word "you" as in "You aren't doing what you need to be doing." Instead, you must consider how to reword things in a helpful light. As in "Is there anything that I or anyone here can do to help?"
I know this sounds really generic in my example, but work ethic today with the younger generation is sketchy. Really sketchy. And even though you're right, and they're wrong, you still have to approach it cautiously.
Sometimes, no matter how much you sugar coat something, a person will take offense. This can, and should be expected. I would recommend a few weeks at least for things to return to normal...although in this case, normal was the problem, so we'd want better than normal, as in him doing his job.
As mentioned previously, his reaction, both immediate and prolonged, will tell you what to do next.
Do I document his activity and report the issue when he will eventually get caught?
If this occurs, which I doubt you should be doing, as it's more of a task for the person's superior, not his coworker, I assume you mean document with tangible proof (internet tracking and login info and whatnot that isn't subjective.)
Do I tell the whole team and let them deal with this character as a collective?
This...can lead to an unhappy workforce. Because then everyone knows about the person, but
generally no one's willing to do anything about it, which only increases the animosity for the person. Pretty soon, everyone's talking about everyone else behind their back.
Do I quietly tell the boss?
Your best option if your first mode of action doesn't produce results within a month or so. And of course, professionally is the way to disclose it, rather than with emotion.