How can I change my personal information on Blizzard account?

Generally the method would be to contact Blizzard's Customer Support Service by telephone. The number should be listed on their website. You will likely also be asked to provide by fax or email at least one form of photo identification. This is the same method used to turn off Parental Controls on your account, or to prove your YOU in the event that Blizzard feels your account has been hacked or otherwise stolen.

Remember, Blizzard employees will NEVER ask you for your password. (something to keep in mind during times such as these)

As a complement for @Ender's answer. Here is a Ticket that I submit to blizzard.

The reply

Greetings!

The first and last name on the account can only be changed by us, in this case I'd highly recommend you contact us for more information directly over the phone.

Hopefully the information will help out, but if you need more assistance, I'd like to reiterate that you can always resubmit a ticket or give us a call directly.

We're here daily between 7am and 8pm Pacific Time, at 1-800-59-BLIZZARD (800-592-5499). Customers in Australia should call 1-800-041-378.