On ones own account, not as part of a general enterprise or venture. Notice the varying accounts (1837, 1841, 1850) of the origin of the phrase.
1812. They forget that [Commodore] Rodgers himself says that he went upon his own hook.Boston-Gazette, Nov. 23.
1836. Did he make these forgeries on his own hook, or at the instigation of the big bug? If the latter, which is most guilty?Phila. Public Ledger, Aug. 24.
1837. The enthusiastic Jerseyman, who, without belonging to either side, was found, at the battle of Monmouth, peppering away from behind a fence at whatever he fancied a foeman,fighting on his own hook entirely.R. M. Bird, Nick of the Woods, ii. 133 (Lond.). (Italics in the original.)
1841. Mr. Weller of Ohio explained that he was precisely in the same position with the old man at Yorktown, viz.: fighting in the political battle on his own hook. Mr. Jenifer said the gentleman had intimated that he was resolved to fight on his own hook; but he would remind him that people were sometimes hung on their own hook.House of Repr., Feb. 15: Cong. Globe, pp. 178, 179.
1842. Mr. Arnold of Tennessee did not pretend to be in the arcana of either party; he fought upon his own hook.The same, Aug. 11: id., p. 80.
1845. The time is fast approaching when we shall have our American Pope, our American Cardinals, and American Catholic everything on our own hook.N.Y. Herald, Oct. (Bartlett).
1846. Now you say these men went out on their own hook; the men who have gone to Oregon must take care of themselves.Mr. Baker of Illinois, House of Repr., Jan. 29: Cong. Globe, p. 278.
1846. General Gaines had raised an army on his own hook. He had called out troops on his own authority.Mr. Archer of Virginia, U.S. Senate, June 5: id., p. 930.
1850. When resistance becomes necessary, North Carolina will act on her own hook.Mr. Stanly of N.C., House of Repr., March 7: id., p. 488.
1850. Mr. Brown of Mississippi said the gentleman from Virginia (Mr. Seddon) reminded him of a valiant youth who, on the day succeeding the battle of New Orleans, was seen busily loading his gun and firing; and upon being asked what he was doing, he replied that he was carrying on the war on his own hook.The same, Sept. 7: id., p. 1775.
1853. Off we started, every man on his own hook, for although we had those among us that afterwards distinguished themselves as officers, yet, with the exception of an old Indian-fighting general, no one assumed any particular command.S. A. Hammett (Philip Paxton), A Stray Yankee in Texas, p. 152.
1866. Attempted to turn a penny and get an honest living by procuring subscribers to various magazines and periodicals, on his own hook.Seba Smith, Way Down East, p. 216.