During the middle ages executionners were a family business. To become an executioner you had to be the son of an executioner. To be fair it wasn't really specific to that profession, most sons took up the job of their parents but when it comes to executioners it was more of an hard rule than an habit of convenience.

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Not to worry, if you weren't the son of one you would most likely not want to be one. I am speaking of medieval France here as I don't know as much about over countries but I suppose things were mostly the same. Executioners were highly ostracized, they were seen as having a sinful professsion and lived isolated from the rest of society. Executionners almost exclusively married with daughters of other executionners for example, which lead them to have consanguinity issues.

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When it comes to the pay they weren't actually paid most of the time, as paying someone for killing is kinda unchristian. Instead executionners, who would be known by the people of their town, simply had a right to take what they needed. For example a baker would always put one of his bread of to the side and upside down, signifying to his patreons that it was reserved for the town's executionner, which for the anecdote lead to the french belief that putting bread upside down was bad luck. So it wasn't very well paid, but it was a very stable situation as executioners will always have what they needed to live.