The cartoon part of the graffito has a different origin. According to Dave Wilton, it is originally British, named Mr Chad, and apparently predates Kilroy by a few years. It commonly appeared with the phrase "Wot, no ____?" underneath, with the blank filled in by whatever was in short supply in Britain at the time--cigarettes, Spam, etc. The Oxford English Dictionary lists Chad's origin as "obscure" but it may have been created by British cartoonist George Edward Chatterton.
Sometime during the war, Chad and Kilroy met and merged, the American phrase appearing under the British drawing.
The combined logo acquired momentum, appearing wherever servicemen travelled, and quickly infected the civilian population. The mania peaked during the war, lingered into the 50s, and then pretty much died out, the joke forgotten as memories of World War faded.
There have been recurrences and imitators. There was a Canadian version named Clem. In the late 60s, there was a version in Los Angeles called Overby. But none of these approached the popularity and ubiquitousness of the original.