What type of cipher or encryption method is this called? Let's say I have a key value of 5.

The message I want to encrypt is "AAAA"

A = 1,
B = 2,
C = 3,
and so on...

the final message is "FLRX"

Each letter will be the value of itself + the value of the previous + the key

1. A = A + previous + key = 1 + 0 + 5 = 6
*6 corresponds to the letter F
2. A = A + previous + key = 1 + 6 + 5 = 12
*12 corresponds to L
3. A = 1 + 12 + 5 = 18
*18 corresponds to R
4. A = 1 + 18 + 5 = 24
*24 corresponds to X


What is this method called? I've been looking at most known ciphers and it doesn't seem to match any. The reason it caught my attention is because the same letter doesn't correspond to the same output much like the majority of the well known ciphers. It is based on the previous letter assignment.
Adding the previous block to the plaintext before encryption is what the CBC mode of operation does. So you're essentially treating the Caesar cipher as a block-cipher and using it in CBC mode.

This adds no security relative to a plain Caesar cipher, since the attacker can simply undo the addition before proceeding to break Caesar as usual.


  the majority of the well known ciphers


That property is shared by substitution ciphers, but not by Vigenere. Modern ciphers don't have any such obvious weaknesses either if you use them correctly.