This was a graffito, of unknown origin but used predominantly by members of the US military and predominantly during WWII. It was often combined with an earlier cartoon image, known as Chad, although the two aren't related apart from that conjunction.

The Chad image, usually with the text 'What, no ...', is sometimes credited to the British cartoonist George Edward Chatterton (a.k.a. Chat), but that is no more certain than the origin of 'Kilroy was here' itself. The Times, in April 1946, pitched it about right when they said:

"Mr. Chad probably went through a number of evolutions at the hands of a vast number of people before reaching the present state and can claim no one man as creator. 'Wot! No father,' it might well complain."

The Kilroy was here graffito was extremely common during the latter part of WWII and into the 1950s and amounted to something of a craze - with it turning up in obscure locations all around the world.

The first reference to 'Kilroy was here' that I can find in print is from US newspaper The Saturday Evening Post, October 1945:

"On the crib hung a hand-lettered sign which asserted: 'Kilroy slept Here'... Wherever he was, Kilroy had been there and left his mark behind: 'Kilroy Was Here', or 'Kilroy Passed Through' or 'You're in the Footsteps of Kilroy'."

The origin isn't known, and there are many suggested derivations, but the phrase is frequently associated with an American serviceman - Francis J. Kilroy, Jr. Two articles that purport to explain the origin, again from US newspapers, are early enough to carry some credence. The first if from the New England paper The Lowell Sun, November 1945 - headed How Kilroy Got There:

The [US] Army public relations office said that a friend of Sergt. Francis J. Kilroy, Jr., of Everett, early in the war wrote on a barracks bulletin board at Boca Raton Army air field in Florida: "Kilroy will be here next week." Kilroy was ill with flu at the time. Later the catchy phrase was picked up by other airmen who changed it to: "Kilroy was here," and scribbled it on air force station walls. Kilroy himself only wrote it a couple of times. By the time Kilroy got overseas, the public relations office said, the thing had gotten out of hand and Kilroy even acquired a cousin. One sign at an Italian base said: "Kilroy's cousin, Corduroy, was here."

A few weeks later, in December 1945, The Nevada State Journal includes Kilroy's own claim to be the original Kilroy who was there:

Now awaiting a discharge at Devis-Monthan field, Tuscon, Ariz., Kilroy informed his parents here that while he was hospitalized earlier in the war a friend scrawled on the bulletin board at a Florida airbase: "Kilroy will be here next week."