I am interested in opening an anonymous offshore bank account but I
need a little direction & guidance. Here are my requirements:

1) It needs to be anonymous i.e. that I wouldn't have to provide
identification to open it.

2) I would need to have access to my account via the internet 

3) It would need to have a routing number and checking number like any
bank here in the U.S.

4) I would like to spend as little as possible, definitely under $1000

If there are any offshore banks that would fit these requirements, I
would accept that as an answer combined with the steps involved in
setting up the account. Also any info about how an off-shore checking
account would differ from a normal U.s. checking account would be
helpful as well.

* I am inclined to tip generously granted the answer exceeds my expectations *

There many sites that relate to opening offshore accounts- this one
seems legititmate (no gurantee).

http://www.offshoresimple.com/offshore_bank_accounts.htm

At the bottom of the page they posted this:
"WARNING:
You must know that the days when you could open a bank account with a
fake driver licence have long gone...
Nowadays, specially after the 11 September events, the banks require a
lot of references for the bank signatories, and due diligence to open
an account.
The MINIMUM requirement is a NOTARIZED passport copy. 
All other banks will also ask for at least one bank reference, plus a
minimum deposit of at least $1000."

Most seem to support internet transactions.  Doing your banking
through a corporate entity created by you may allow you to bypass
identifying yourself- but i highly doubt it.

Good luck.

Before you open that account, you may want to at least read the
Patriot Act.  It could have a negative impact on your desire to
expatriate your funds.  Repatriating them even years later is no
picnic.