The complication in trying to identify the origin of "breaking bad" is that "break bad[ly]" has a number of figurative uses (not involving the physical splitting, sundering, or shattering of something) that antedate by many decades the probable slang that the TV show title Breaking Bad alludes to.
'Breaking bad' on the track
One old figurative use relates to horse racing, where the sense of break seems to be something like "falter" and may be related to definition 11(a) of the transitive verb break in Merriam-Webster's Eleventh Collegiate Dictionary (2003):
11 a : to check the sped, force, or intensity {the bushes will break his fall} {without breaking her stride}
Examples of this sense of the"break bad[ly]" date at least to the middle of the nineteenth century. From "Sporting Intelligence," in the [San Francisco] Daily Alta California (May 9, 1853):
SECOND HEAT.—On the first start they got the word. New York led off and kept his position to the 1st quarter pole, when he skipped, and Dominick, who had been gallantly waiting upon him, came up, collared him and took the track. The pace was too killing to last long. Before reaching the half mile pole Dominick broke bad, when New York and Creeper came up and passed him to a "dead stand." Here there was some beautiful playing between New York and Creeper—now New York, then Creeper, then you could have covered them both with a blanket; in the meantime Dominick had regained his foot, and was "bumming" along at an awful pace. ...
And from "The State Fair," again in the [San Francisco] Daily Alta California (September 24, 1880):
The fourth heat was well contested between "Button" and "Annie," the latter again breaking bad on the back-stretch, but coming up the stretch with such speed that she went under the wire a length ahead of "Button," but the Judge* declared it a dead heat, in consequence of "Annie" not having been got on her feet, after breaking, as soon as the rules require.
'Breaking bad[ly]' in the market
Another old figurative use—almost certainly the one intended in the 1919 citation noted in Third News's answer—relates to the Eleventh Collegiate's definition 12 of the transitive verb break:
12 : to cause a sudden significant decrease in the price, value, or volume of {news likely to break the market sharply}
This usage dates back at least to the 1880s. From "The Globe Telescope: As It Casts Light on the Chicago Markets" in the [St. Paul, Minnesota] Daily Globe (February 20, 1881):
Chicago, Feb. 19.—The speculative interest here has been centered in provisions to-day, that market breaking badly on a report that the French government prohibits the importation of American meats.
And from "Excitement in the Stock Market," in the Lancaster [Pennsylvania] Daily Intelligencer (July 3, 1881):
Philadelphia, June 2.—12:50 P. M.—The stock market shows increasing excitement and prices on some of the stocks are breaking badly. The stock board has passed a resolution to continue in session during the afternoon. Northern Pacific common is quoted at 40.
The market excitement in this case was caused by the assassination of the U.S. president, James Garfield.
Dictionary coverage of other slang senses of 'break bad'
The TV show sense of "breaking bad," in contrast to either of the two preceding figurative senses, seems to involve the following meaning, given in J.E. Lighter, The Random House Historical Dictionary of American Slang (1994):
break bad Black E[nglish] to become aggressive or angry. [First three cited instances:] 1972 in W. King Black Anthol. 142: He ain' do nothin', though, did he....Long as he ain' break bad and do nothin', it don't even count. ca 1974 in J.L. Gwaltney Drylongso 20: Those paddies...told [the Czechs] to break bad with the Bear to begin with. ca 1979 in J.L. Gwaltney Drylongso 146: She would break real bad with her daddy and and stomp her foot and just get all out of hand.
As Joe Cervone's answer points out, "break Bad" in the sense that Lighter identifies also appears in Claude Brown, Manchild in the Promised Land (1965):
Down home, when they went to town, all the niggers would just break bad, so it seemed. Everybody just seemed to let out all their hostilities on everybody else. Maybe they were hoping that they could get their throat cut. Perhaps if a person was lucky enough to get his throat cut, he'd be free from the fields. On the other hand, if someone was lucky enough to cut somebody else's throat, he'd done the guy a favor, because he'd freed him.
The odd thing here is that two significant glossaries of African American slang that were (like Lighter's dictionary) published in 1994 have no information on "break bad." The closest we get in Clarence Major, Juba to Jive: A Dictionary of African-American Slang (1994) is this entry for breaking:
Breaking adj. (1980s–1990s) to be obsessive about something or to go to extremes.
And in Geneva Smitherman, Black Talk: Words and Phrases from the Hood to the Amen Corner (1994) the most relevant entry—distant though it is—is this:
BREAK ON SOMEBODY 1) To talk negatively about somebody. 2) To embarrass a person in front of others.
I have no idea why neither Major nor Smitherman addresses "break bad"; the commonsense inference is that usage of the expression was not general in Black English, despite the instance from Claude Brown's classic 1965 novel and the several instances that Lighter cites from the 1970s.
In a newspaper search, the earliest match I could find for "breaking bad" in the period 1960–1980 was from a "Free Unclassifieds" notice in the [Washington, D.C.] American Eagle (September 22, 1978):
DEAR SP - SORRY Ive been indisposed. Hang tight, Baby, I'll be breaking' bad soon. Love, BM.
The American Eagle is the student newspaper of American University, and the sense of the expression here appears to be definition 2 in Jonathon Green, Cassell's Dictionary of Slang (cited in ermanen's original question): "2 {1980s} (US campus) to perform well."
Interestingly, Jonathon Green, Chambers Slang Dictionary (2008) adds a much earlier sense of "break bad" to the two that he had reported in Cassell's Dictionary of Slang:
break bad v. 1 {1910s} to have a mental breakdown 2 {1960s+} (US black) to become angry or aggressive. 3 {1980s} (US campus) to perform well.
In his introduction to Chambers Slang Dictionary, Green notes that "The + sign [attached to a usage date] indicates that a term is still in use." Obsolete terms that lasted for across more than one decade are identified with an inclusive range such as "1920s–1940s," so the designation "1910s" for "to have a mental breakdown" indicates that this sense of "break bad" did not outlive the named decade; likewise, the "to perform well" sense of the term did not persist beyond the 1980s, according to Green.
Conclusions
From the foregoing evidence, it seems to me that the TV show's immediate source for "breaking bad" was the anger/aggression sense of the term identified by Lighter and Green as having originated in the 1960s or 1970s in US Black English.
Although this meaning might in its turn have been influenced by figurative or slang senses of "break bad" used in horse racing, financial markets, or insane asylums, the sheer number of meanings of break as a verb renders a clear path to any of them exceedingly difficult to identify. (The Eleventh Collegiate provides 43 definitions for break as a transitive verb and 30 definitions for break as an intransitive verb.)
On the other hand, it is possible that the immediate inspiration for the TV expression might come from yet another strand of colloquial usage. Consider this instance from "Farmers, Law Officers Say Attacks on Sheep Have No Simple Solutions" in the [Monterey, Virginia] Recorder (June 2, 1988):
Wright said most hunters are responsible citizens and keep a watch on their dogs. "I don't want to see a leash law, because the same people who use their dogs for hunting won't let them run in sheep or cows."
Wright pointed out that some I hunters have even killed their own dogs they found attacking sheep.
Shultz, who is a hunter, said occasionally a hunting dog will "break bad on you and get in sheep. But," he offered as an insight, "if a man cares very much for his dogs, he's going to keep them in line."
The sense of "break bad" here seems to be something like "fail to obey instructions or training and instead do something harmful, destructive, or otherwise intolerable"—or, in short, "go rogue." From my understanding of the TV show, that meaning might be somewhat applicable to the show's main character.
