By 1999 there were plenty of computer-based security laws on the books. In general, the kinds of things Neo could have been doing on a computer to break the law fall into three legal areas:
Copyright infringement: The DMCA went into effect in 1998, but the US had been trying to enforce existing copyright law on computer piracy for much longer then that. Remember, the Internet went commercial in 1993, and exploded rapidly. DVDs were out by then, and piracy of movie, television, software, music, etc. was becoming a hot-button issue. Napster started that year, so it wasn't a big factor yet, meaning most piracy was being done by a small number of individuals on a larger scale.
Intrusion and Theft: The Secret Service had been investigating people who used computers "without authorization" dating back to the 80s. There were a couple of very high-profile investigations during that period. Again, they were mostly trying to enforce existing laws, like trespass, theft, and property desctruction, as applied to online actions. Title 18, Sections 1029+ went into effect in 1994, making the unauthorized use of access devices and computers a federal crime in its own right; it also applied the "wire fraud" laws originally used for telephones to internet and other network-based access.
Pornography. The CDA is the obvious one here, though it may have been dismissed by that point. As far as actual illegal activity here, lets just assume Neo had nothing to do with that stuff.
When Smith says Neo broke "every crime we have a law for", if he just mean specifically computer-related laws, it wouldn't have taken a lot. The way many of these laws are written, a single "hacking" session -- say, breaking into a software company's servers, downloading a game, breaking the copy protection, and distributing it -- would hit about 80% of them. If he means every law that the Secret Service tried to apply to computers, then Neo would have been a pretty busy person.
As for how good a hacker he was -- well, he was good enough to grab the attention of the government (and Morpheus), but not good enough to avoid getting caught, so...