Black Hat hacking differs from White Hat hacking only with the way you are using the results of this hacking. You can't be Black Hat hacker when hacking into your own website.
If your site is located on a standalone server, you could use port scanner to define open ports on that server. Then, using port scanner you could try guessing exact services that are running on those ports. After you get that information, you could start using any exploits that are designed for those services. I suggest using NMap as a port scanner. Try reading some articles or get info on forums on how to use exploits.
If your site does not run on standalone server, do not try scanning ports on that server or using any exploits designed to take over that server. You may have serious problems if you start doing it.
You could also test your site for possible SQL injections or XSS vulnerabilities. If that site belongs to you, this is completely legal to test it for these vulnerabilities. Try getting more information about that on forums and communities, even if this information is designed for Black Hat hacking.
If any of the tests gives you positive results, you should consider modifying the code of that section to make it more secure.
If the site does not completely belong to you, don't try running any tests on it, as it may be considered a hacking attempt. For example, livejournal, myspace or blogspot are well-defended against attacks.