It's a story that gets mentioned in various books on North Korea, some accept it as fact, others as hearsay. What's the real deal?
This dovetails into a recent thread that the mods had to nuke because it was against the rules, and I am shamelessly going to amend my response for that here.
Short answer: We'll probably never know for sure, but it's plausible.
Long answer: Given what we know of the history behind North Korean propaganda, how the government has handled biographical material of the Kims, and how Kim Jong-il sought to manipulate public opinion to bolster his odds of succession, the story is within the realm of possibility. Either way, the whole mess is a good guide en miniature to the extreme difficulty of assembling North Korean history.
So what happened? The original story comes to us via a North Korean official who defected in 1960 (more on this in a bit, because how we got the story is important). Most North Koreans, even in the upper tier of its society, are never told that Kim Jong-il wasn't born in North Korea at all. He was born in a Soviet army camp in Siberia in 1942, and he was followed by a little brother in 1944. This little brother's existence is also unknown to most North Koreans, for reasons that will become apparent shortly.
Russian nicknames (also something the North Korean government doesn't advertise): The two boys were nicknamed Yura and Shura respectively (Russian nicknames for Yuri and Alexander -- and in fact, we only know the younger brother's nickname for sure), and they accompanied their parents back to Korea after World War II ended and the Soviet army descended into the peninsula. The story that concerns us today comes from 1947 or 1948, when Jong-il would have been 5-6 years old and his brother 3-4. They were said to have been playing in a pond in Pyongyang, and as his younger brother repeatedly tried to climb out of the pond, Jong-il pushed him back in. Shura eventually tottered under the water after another push, was too exhausted to pull himself out, and drowned. Their parents were too late upon the scene to rescue him, shouted at Jong-il for an explanation, and didn't receive one.
So. If the story is true, then Kim Jong-il did in fact kill his younger brother (or was at least responsible for his death) at a very young age.
Why it might not be true: Trying to assemble an accurate narrative about events within (or concerning) North Korea is difficult even under the best of circumstances, because NK's government has a very opportunistic approach to its own history. Events that support the narrative of Kim il-Sung, and secondarily Kim Jong-il, as the great saviors of their nation, are exalted and embellished, if they aren't being made up entirely. Events that do not support this narrative are downplayed at best and erased at worst. That's bad on its own, but the acrimonious relationship between North and South Korea during the Cold War added another layer of complexity. With anything that was filtered through the South Korean government at the time, it's prudent to ask whether they were putting their own spin on the information, or simply not publicizing anything that might have countered it.
The defector who provided the story arrived in SK about 7 years after the Korean War's ceasefire. This was a time when when passions and propaganda were running high on both sides of the border, so there are a few possibilities that we try to consider:
If you want to tear your hair out by this point, that is a completely rational reaction to the experience of studying North Korean history.
A quick note on the story and why it could circulate in NK without everyone involved getting thrown in a prison camp: People are used to the assumption that "Anything bad said about a Kim family member within North Korea" = "One-way ticket to a prison camp for you and your entire family," so it's important to remember the time period. Dynastic succession wasn't a "thing" in Communist governments (Marx had inveighed against the whole notion of inherited power), and people who worked in the early North Korean government would not necessarily have thought that Kim il-Sung's job would pass to a son. Kim Jong-il wasn't yet part of NK's government when the official defected, and even after he became part of the government, he wasn't universally liked. So the official may not have attached the kind of weighty importance to the story that we're inclined to do today, because Jong-il was just a pain-in-the-ass college kid at the time and not necessarily a figure of great destiny in North Korean culture.
Also, the first of NK's prison camps probably dates to the late 1950s at the earliest, and the massive security and police apparatus that developed to move people into the camp system wasn't fully in place.
Why it might actually be true: Historians tend to entertain the defector's story as a genuine possibility, even if they don't necessarily believe it means Kim Jong-il was a childhood sociopath. Thanks to Soviet army records and pictures from the era, we know that Shura did in fact exist, and that his age would have made either the villagers' or the official's stories plausible. However, he is curiously absent from most official North Korean histories even though the circumstances of his death could have helped Kim Jong-il's cause. When the defector's story started to circulate in South Korea, the North Korean state's only response to it was silence. Not a denial, not a correction -- just silence. That still doesn't make it true, but it's something to consider.
Kim Jong-il was active in North Korea's government from his early 20s onwards, and he started his career by controlling the country's art and literature scene (he was mostly interested in/famous for the country's movie scene, but it wasn't the only thing he did). Even from an early age, he had absolute control over his personal narrative within North Korea, and he was big on playing up anything that could win him additional sympathy or support from the North Korean populace (e.g., his mother's early death in 1949, his respect and support for his father's beliefs, etc.).
Why? Because his position as his father's successor was not (we think) guaranteed until much later. He had an advantage as his father's eldest son, but his step-siblings were born more than a decade after he was, in a time when Kim il-Sung was firmly established as North Korea's absolute ruler and could afford to spend more time with them. Kim Pyong-il, who was Kim il-Sung's son by the favored wife, Kim Song-ae, was said to be a particular favorite and the most likely rival for the top job. While Kim Jong-il was working his way up the government ladder and currying favor with Dad, he saw Pyong-il as a threat to his interests and worked hard to eliminate or minimize it. Arguably the most effective way he did this was by controlling and manipulating the information given to the public on the Kim family.
So of the histories that Jong-il approved and published in the 1960s, none of them make any reference to Shura -- it's as if he never existed at all -- and none of them acknowledge that Kim Jong-il had step-siblings. North Koreans are instead presented with a very sanitized story in which Kim il-Sung's only son was:
If Shura had died in a genuine accident or in something completely unrelated to Jong-il, it seems reasonable to assume that Jong-il would have played up the tragedy of his brother's death in much the same way that he played up their mother's early death. Instead, silence.
I've seen some material elsewhere claiming that later North Korean histories do acknowledge Shura (though not the rumor of Kim Jong-il's involvement in his death), but I haven't personally confirmed it yet. One of the most unfortunate and eerie features of North Korean history is that, in marked contrast to history elsewhere, it changes.