Origin of "I kid you not"

I found a print reference of the phrase as early as 1948 in the International Stereotypers' and Electrotypers' Union Journal, Volume 43:

Boy!, Oh Boy! I said I asked for a headache when I volunteered for this job, and I kid you not when I repeat it.

But it seems it was given a much wider audience as a phrase spoken several times by Lieutenant Commander Philip Francis Queeg in Herman Wouk's 1951 Pulitzer Prize winning novel, The Caine Mutiny:

That's the Navy for you. Pass the buck and get a receipt. Act at discretion, hey? Well that's exactly what I'm going to do, and I kid you not.

This link seems ambiguous about the origin, but seems to be popularized by TV show host Jack Parr

: : I KID YOU NOT - Catchphrase used by Jack Paar. Paar, host of the Tonight Show from 1957 to 1962, 'invented the talk-show format as we know it: the ability to sit down and make small talk big,' said Merv Griffin. 'Even youngsters sent to bed before Mr. Paar came on parroted his jaunty catchphrase, 'I kid you not.' From "He invented late-night talk, then walked away," an article in the Herald-Leader, Lexington, Ky., January 28, 2004.

Grammatically both your versions are correct - "kid you" and "kid with you"

Your wording of the question suggests someone is upset about a joke you've played on them, so it's more common to say

"Just kidding" rather than pose it as a question.

I always thought this phrase originated from young goats. Young goats are called, kids. These creatures are very playful. A joke is playful and fun, as are kids. Hence, "kidding" equates to the act of being playful.

Google books shows that

I'm not kidding,

used in the same sense as it is in America today, appears in the British novel A Daughter of the Philistines by Leonard Merrick (1897) and in the British novel The Story of the Amulet by Edith Nesbit (1906). All Jack Paar did was invert the word order to get I kid you not.

Merriam-Webster says that kid (used in this way) is a transitive verb, so the correct American usage is

I'm kidding you.

or

I'm not kidding.

Because the verb is transitive,

Can't I kid with you.

would be incorrect because of the with (and it definitely sounds wrong to me). On the other hand, you can say

Can't I kid around with you.

According to the other answers, the expression seems to have died out in Britain.