On the iPhone, go into voicemail and tap "Greeting". Record a few seconds of something and tap stop. Before you tap save, use iFuntastic or SFTP to navigate to /var/root/Library/Voicemail on your iPhone. You'll see a file called "Greeting. amr". This is the audio you just recorded. Delete that file and replace it with your custom message which should then be renamed to "Greeting.amr". Tap "Play" on the iPhone to verify your message works. If it plays, tap "Save" and your custom message will be uploaded to AT&T's servers.
By default, the telephone number is the one assigned to you by your organization, but it could also be a cell phone number or email address. In that case, voice mail would be played back through your email Inbox.

You can play a voice message at different speeds and pause/resume the message, using desktop client visual voicemail, or by dialing into their voicemail. Pressing a certain key on your phone controls how you can hear the voicemail: Read your voicemail transcription
Recording the message yourself doesn’t mean it can’t be done professionally. You can still get studio time with an experienced audio engineer. Studio time can cost $60-$100 per hour, which may or may not come with an engineer. If you need to hire one, their rates are usually $100-$300 an hour. Mind you, their feedback, professional experience, and editing capabilities are typically well worth the extra expense.