The phrase a chip on one’s shoulder means a challenging or belligerent attitude.

In A Concise Dictionary of Phrase and Fable (1993), B. A. Phythian explains:

There is an unusual degree of unanimity about the provenance of have a chip on one’s shoulder (bear a grudge; behave anti-socially). Unlikely as it may seem, the reference is to a custom originating in the USA, but also known in Canada, in which a person who was looking for a fight carried a chip of wood on his shoulder and invited people to knock it off; anyone who did so was agreeing to fight. Perhaps the custom made better sense in pioneering days when chips of wood were litter as common as pieces of paper today, and fighting for its own sake was equally common.

North American newspapers mentioned this custom from 1830 onward. For example, the following is from A Roman Sketch, a humoristic story set in ancient Rome “after the assassination of Julius Cæsar”, published in The Picayune (New Orleans, Louisiana) of 22^nd February 1837; Cassius, one of the leaders of the conspiracy to assassinate Caesar, wakes up Brutus, who also played a leading role in the conspiracy, by tickling him in the ear with a long straw that he has drawn from the bed:

‘What do you mean, Mr. Cassius?’ said he [= Brutus], doubling up his fist; and squaring to his compeer, ‘you ought to be ashamed of yourself.’
‘Give us none of your jaw!’ exclaimed Cassius, who was a remarkable touchy man. ‘But if you want any thing of me, just knock this chip off my shoulder,’—and at the same time he placed a piece of bark there.

The American author James Kirke Paulding (1778-1860) had referred to the custom in Letters from the South, written during an excursion in the summer of 1816 (New York, 1817):

I must not forget to tell you that the only vestige of ancient chivalry I have seen in all Virginia, occurred at Martinsburg. The day being warm, we were sitting, probably to the number of twenty, on benches, at the shady side of the hotel, fronting on one of the principal streets, when a man rode furiously by on horseback, and swore “he’d be d——d if he could not ‘lick’ any man who dared to crook his elbow at him.” This, it seems, is equivalent to throwing the glove in days of yore, or to the boyish custom of knocking a chip off the shoulder; but, alas! well was it said by Neddy Burke, the days of chivalry are gone,—and may they never return, say I. Instead of ten thousand fists leaping from the pockets of the supine spectators of this magnanimous outrage, they affected to take no notice of it; and, by heaven! not one accepted the challenge! Degenerate days!—and how unlike the fabled times, when such a gallant raid as this would, according to Dr. Morse, have cost many an eye, and many a bloody nose.

The following, from a story published in the Carolina Observer (Fayetteville, North Carolina) of 25^th November 1830, insists on the fact that the person who knocked the chip off the challenger’s shoulder accepted the consequences of the ensuing fight:

“Phineas L. Tracy is on the look out for yon [= you],” said some one. “Let him come, said I, “I can thrash him any day—I have done it before in Washington, and I’ll do it again.” Then there was a report that Fred Whittlesey was going to waylay me. “He waylay me, said I,” the mean sneaking fellow—I am only afraid that he will sue me for damages. 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.”

The earliest figurative use of the phrase that I have found is from The Sun (Baltimore, Maryland) of 27^th April 1855:

The executive government have decided one question of policy, to wit: that a pretension by Spain to overhaul our vessels, on the high seas, in search of fillibusters [sic] or for any other object, is not to be admitted—and that, on the contrary, the practice founded on such a pretension, must be discontinued or promptly, and on the spot, resented.
Commodore McCauley, in pursuance of this decision, goes out with orders for its practical enforcement. Inasmuch as the orders are prospective, and apply to no bygone case, the commodore must await a chance for their application.—The effect of the orders, when known to the Cuban authorities and the home government, must be to cause a relinquishment by Spain both of the practice and pretension on which it is founded; or to afford her an opportunity and a pretext for a collision. The issue is proposed by this government, and it remains for that government to accept it. Any American merchant vessel may, by an understanding between the parties concerned, be made the instrument for the experiment—the chip on the shoulder—whereupon the contemplated ‘duello’ between the San Jacinto* and the Spanish cruizer [sic] may occur.
I protest in the name of common sense, and the common sentiment of the country, against the supposition that Commodore McCauley is to witness an attack on an American merchant vessel, see her fired into and fired at, or sunk, or captured before he is to interpose his protection.

(* USS San Jacinto: a frigate in the United States Navy)