Report or go after the hacker legally?

Hacking is a violation of the terms of service for any legitimate webhost or ISP. If you can prove conclusively that someone is using a particular IP address for hacking (or spamming, too), you could report the incident to the webhost or ISP in hopes that they might shut the perpetrator down. The contact email is often abuse@ the company.

However, your chances of getting anywhere with this aren't very great. Even if you succeed, it's a drop in the bucket. Although you might feel as though you are in a battle of wits with a wily adversary, it is thousands of times more likely that you were hit by an automated drive-by attack that is playing a percentage game, with malicious requests being launched against millions of websites, from hundreds of malicious servers. If one is shut down, it's just a cost of doing business for them.

It is a more worthwhile use of your time to do everything you can to protect your site from all hackers, regardless of who they are, and understand that there will be a constant flood of attacks against your site.