Does a stream cipher require a KDF when it's key is variable length? RC4 has a variable length key. Is it still necessary to use a key derivation function, such as PBKDF2, to initialize it?

Would using a single character password weaken it's output when compared to a block cipher using, for example, the SHA-256 hash of that same single character password as the key?
One problem with RC4 is that, while it does take a variable length input (up to 256 bytes), it's known not to be great at mixing those bytes together.  Specifically, we see correlations between the RC4 key and the RC4 output stream.

My first recommendation to you would be to use something other than RC4.  About the only advantage RC4 has over most other ciphers is that it is simple enough to be memorizable; for example, if you are going someplace where you don't want to carry any programs with you, if you can get access to anything programmable, you can type in an RC4 program, and have a good chance at having it work properly.  If you don't care about that (and you probably don't), then there are better options available.

My second recommendation (assuming that you ignore the first) is that you ought to obscure the relationship between your password and the RC4 key (as RC4 doesn't do a great job of obscuring the relationship between the RC4 key and the keystream).  Using a KDF to process the password would work; at the very least, you should use a hash function.  Now, I don't know whether the weaknesses in the RC4 key setup logic would be a concern to you; I also don't know that it's not; covering that potential weakness would appear to be a prudent step.

As for a single character password, well, that's going to be weak, whether you use a block cipher or RC4....