Karl and Alex are part of a younger generation of Others who don't like the way their elders are running things.
It is likely that Karl and Alex are the only teenage Others, since the Incident made it impossible for babies conceived on the island to be carried to term. Unless there were more frequent instances of pregnant women or young children arriving on the island, then there would be much fewer Others under the age of 27. This could have isolated them and account for the fact that Karl is at odds with the Others.
Karl is either a doctor in training or an aide to those who are doctors. He developed and brought Ben's X-rays to Juliet.
The brainwashing he was being tormented with in Room 23 was meant to cause him to turn against Alex and/or the castaways.
Karl was kidnapped as a child. His real parents are dead, and he was given to an Other to be raised, much like Alex was.
His father is Goodwin. Karl says that he and Alex used to lay in his backyard and name the stars. Therefore, Karl is most likely the son of one of the Others. Goodwin is a likely candidate, since he is about the right age to have a teenage son, and with him being dead, there is no longer anyone looking out for Karl.
But if his father is Goodwin, his mother would be Harper, which means there is someone looking out for him.
Maybe, but it's also possible for the mother to be someone else Goodwin got involved with beforehand. Perhaps she tried to have another baby post-Incident but died?
Since Ben indicated that his disapproval of Karl and Alex's relationship was solely from the fear that Karl would get Alex pregnant, that could mean that he has already slept with her.
Ben also says to him at some point, "If you're going to sleep with my daughter...." which indicates they are already involved physically.
Since it appears that children conceived on the island can't be carried to term, Karl was probably conceived and maybe even born in the outside world, then brought to the island as a small child or while still in his mother's womb. His biological parents were DHARMA researchers who died in the Purge, but he was 'adopted' and raised by Others in much the same way Alex was.
Karl could have been in a similar situation to Ethan Rom, born on the Island or came at a young age, and was conditioned into joining the Others. Karl is much younger than Ethan however, so this must have happened later on in the timeline.
Karl also could have been born in a similar situation as Aaron, Alex and Jacob/The Man in Black: born on the island to a pregnant woman who was stranded there. Karl doesn't seem to think there is anything strange about their world, which indicates he was there all his life.
The Others are also not above kidnapping children. Since it became obvious after the Incident that their population was endangered (and recruiting newcomers from the outside risked exposing their world), it is possible they kidnapped children or bought trafficked babies on the black market.
In "A Tale of Two Cities", Karl told Sawyer to run in a different direction as a distraction for his own escape.
One could assume, from his confusion over Sawyer's Brady Bunch comment, that Karl may have been born on the Island (if not brought there as a small child, not unlike Zack from the tail section).
Of course, not everyone knows the Brady Bunch.
Every American (with the exception of religious sects like that Amish) knows what the Brady Bunch is.
Karl's last name may be a humorous reference to the Brady Bunch, since that is the last name of the girls from before Carol married Mike.
Aaron is the only person that was definitely born on the Island and that his mother was pregnant before arriving on the island. Karl was not conceived on the island. He either was brought to the Island after his birth or his mother came to the Island while already pregnant.
Alex and Ethan were also definitely born on the Island. Karl could have been born on the Island if his mother had arrived already pregnant, like Alex's mother.
Karl is of a German-speaking descent. This is indicated by the fact that Karl with a "K" is much more common in these countries than elsewhere. He was either born in that region or has a parent of German-speaking origin. This would also explain the lack of knowledge of The Brady Bunch, in the German version known under Drei Mädchen und drei Jungen.
That's not strictly true. While the alternative German spelling makes it more likely that he is of German descent, it doesn't definitively make it so.
Karl could not have been the German tailender or any other survivor of Oceanic 815. He developed Ben's X-rays for Juliet prior to the crash. Additionally, Karl may not have been his real name, if he was kidnapped as a baby or brought to the island under any other nefarious circumstance.