Origin
The phrase originated in the United States and is from the 19th century. People who were looking for a physical fight would walk around with a chip of wood on their shoulders, daring anyone nearby to try and knock it off. To accept the proposed challenge, someone would have to knock the chip of wood off his shoulders. On May 20, 1830, A New York newspaper, Long Island Telegraph, printed:
"When two churlish boys were determined to fight, a chip would be placed on the shoulder of one, and the other demanded to knock it off at his peril."
It appeared once more in New York, on December of the same year, in an issue of The Onondaga Standard:
"Oh! if I only could get him to knock a chip off my shoulder, and so get round the law, I would give him one of the soundest thrashings he ever had.”
Examples of the Phrase Being Used
"Why are you still mad at me? You got a chip on your shoulder or something, pal?"
"You seem to have a chip on your shoulder for that fellow, Louis."
"Betrayed by a friend in the past, I've had a chip on my shoulder ever since. I just can't let it go."
Note: For most old phrases, sayings, and idioms, finding the exact person or date in which a phrase was originated is impossible. What's provided is the earliest known date the phrase or saying was being used in.
Usually the phrase is a quote from an old newspaper, book, poem, play, etc. If it is already being used by a newspaper though, then the saying must already be common knowledge among the people of that time. This means the origin of the saying is older.
For instance, if someone or something from 1660 is being quoted using a particular phrase or saying, you can assume that the saying must be older, as it's already popular and being commonly used in 1660. How far back varies from phrase to phrase.
There are some cases though where determining the origin from an exact person is possible, however these are not common. In short, most of the origins for these sayings give you a date so you can get a rough idea on when these phrases were being used.