Background on 'bucking'
"Bucking" in the sense of "avidly pursuing" seems to have its origins in U.S. military slang, but it has much broader application today, as Kristina Lopez notes in her answer. The earliest instance of the word used in this sense, according to J.E. Lighter, Random House Historical Dictionary of American Slang (1993), is from 1881—and that instance doesn't seem to be military-related, although military use is responsible for the term's widespread adoption in the twentieth century:
buck v ... 3. Esp Army. to strive fervently (for a position, promotion, advantage, etc.), esp. by currying favor.—often used with elliptical complement, e.g., buck for corporal 'strive for a corporal's rank'. {The orig. army phr. seems to have been buck for orderly, explained in 1909 quots.; extended use, often ironic, became common *ca*1940.}
[First three cited examples:] 1881 in [Mitford Mathews, A Dictionary of Americanisms on Historical Principles (1951)] 201: I was bucking very strong for the job. 1900 McManus Soldier Life 104: Many of the old-time orderly-buckers refused to buck at guard-mount, preferring to take their chances on being assigned to the favorite post down below the old Malate church. 1907 Bush Enlisted Man 30: The first sergeant told me to go in and "buck"for orderly. He said the company usually out an orderly "bucker" on their guard detail and generally managed to corral the prize.
The first of these examples is from Henry Hayes & Charles Hayes, A Complete History of the Trial of Guiteau, Assassin of President Garfield (1882):
On cross-examination [of a "colored hack driver, Aquilla Barton"] as to the prisoner's [that is, Guiteau's] appearance at the time of the shooting, the answer was given that the prisoner was not excited.
A. Were you excited? A. Not at all; I was not, but I was bucking very strong for the job; he was a good deal in his senses; he was "flesher" than he is now. [Laughter.]
Prisoner, in a humorous manner: I may state here that I had the first square meal to-day I have had since the 2d of July. [Laughter.]
The original recorded instance of bucking to mean "strive fervently for [a job]" thus appears to involve a Washington, D.C., hack driver's slang usage at the murder trial of Charles Guiteau. A similar instance, also from Washington, D.C., and involving "a colored hackman" appears in "Fought in Front of the Capitol," in the Washington [D.C.] Times (March 5, 1895):
Thomas R. Jamison and Charles Braxton, hackmen, celebrated the adjournment of Congress yesterday by engaging in a lively affray in front of the west side of the Capitol. They were going at each other "hammer and tongs" when Policeman Mendenhall, of the Sixth precinct, appeared and placed both men under arrest. Marshall Brown, a colored hackman, was arrested at the same place by Policeman Whalen while "bucking" for trade or soliciting passengers in violation of law.
The fact that two instances from Washington, D.C., mention bucking in the context of a hack driver trying to get prospective passengers to hire him (in the latter case in violation of a local ordinance against such solicitation) raises the possibility that "bucking" for trade was highly localized slang for "solicitation," and not a precursor for bucking in the military sense of striving for some position or honor.
One of the 1909 instances mentioned in Lighter is actually from two years earlier—from Hearings Before the Committee on Military Affairs, United States Senate, Concerning the Affray at Brownsville, Texas, on the Night of August 13 and 14, 1906, volume 2 (1907):
By Senator WARNER: Q. Explain that—what is meant by "bucking for orderly?"—A. The cleanest man on guard when they mount guard in the morning is picked out by the adjutant for the commanding officer's orderly. He does not have to go on guard duty, and is only on duty practically from about 10 in the morning until about 12, and from 2 till about half past 4, and then he is allowed to go to his quarters. It is something they all desire, and the men will clean up, and as the competition becomes closer and closer they get down to even quite small details, and the brass screw that is in the head of the bayonet, that ill be polished, and they will get down to the point, some of these men, where they will even polish their cartridges.
But "bucking for orderly" is explained in detail in the [New York] Sun (April 11, 1897) in an article of that title:
BUCKING FOR ORDERLY
THE CEREMONY OF GETTING A SOFT BERTH AT AN ARMY POST.
The Rivalry Is Confined to the Spick and Span Men Who Seek Light Duty in the Immediate Service of the Commanding Officer, but the Whole Post Is Intensely Interested.
Washington, April 10.—"Bucking for orderly" is a struggle that takes place at every military post in America every day in the year at guard mount. It is a struggle that inspires the bosoms of the enlisted men with ambition, envy or rage, according to the contest's outcome, and at the isolated posts it gives the men something to live for between pay days. ...
Bucking for orderly is the strife among the men detailed for guard duty to be chosen orderly for the commanding officer by the Adjutant at guard mount. Out of every guard mounted a man is selected by the Adjutant to report to the "beak" or the "old man"—the commanding officer of the post—for messenger duty. ...
Mathews has another interesting definition of buck that might be related to the one in question:
buck v. ... 2. intr. ... b. To buck up to, to "shine up to" a girl, to seek to make a good impression upon. Colloq. 1832 Polit. Examiner (Shelbyville, Ky.) 8 Dec. 4/1, I seed her at church one day fixed up kinder pretty snug, do ... darn my seelskin pumps if I dont buck up to her next Fust day. 1868 PAULDING Book of Vagaries 265 Single gentlemen ... should beware how they 'buck up' to widows.
The shared element here is "seek to make a good impression upon," but no authority I'm aware of has suggested that the "shine up to" meaning from 1832 is directly responsible for the "strive fervently for a position" meaning from 1881.
Answers to the the posted questions
1. Is this one of those annoying single-use phrases?
No, it isn't. The phrase seems not to have originated in military use, although that is certainly where it became widespread. Today, the most common phrase using bucking in the relevant sense appears to be "bucking for a promotion"—which may refer to an upgrade in military rank or to an advance in civilian employment. Other phrases that a Google Books search finds include "bucking for a raise" (where buck for a raise is defined as "To be very determined about getting a salary increase)," "bucking for a job," "bucking for a prize," "bucking for a halo," "bucking for a serious bonus," and "bucking for a spot on Queer Eye for the Straight Guy." There are many nonmilitary instances.
2. What is the origin?
The answer is uncertain. In military use, the earliest phrase associated with this type of bucking was "bucking for orderly"—trying to look sharp in hopes of being selected to fill the easy job of commanding officer's orderly for the day. The hack driver's use of bucking in the sense of striving to be assigned a desirable task 16 years before the first known military use clouds the picture. The 1895 example of "bucking for trade" suggests, however, that "bucking" as "soliciting" may have been local Washington, D.C., slang among cabbies and policemen, and not a direct predecessor of "bucking" in the "bucking for orderly" sense. The use of "bucking up to [a person]" in a somewhat similar sense to "bucking for [a position]"—namely, to seek to make a good impression—makes the situation even foggier.
3. What other phrase or word could you use to describe Klinger's actions?
There are many alternative ways of expressing the idea of trying to achieve a particular result through assiduous effort: "aiming [or shooting or gunning] for," "scheming for," "making a play for," "intriguing for," "going all out for," "concocting elaborate designs for," etc. All of them have slightly different meanings, though, and none of them is an exact replacement for "bucking for."