In "A Good Man is Hard to Find," what has the Misfit’s life been like? Why was he put in jail? He says he kills for pleasure. Is there another reason?

The Misfit claims that he was raised well and that his life turned when he was put in prison for the murder of his father. However, he also remembers his father dying 1918. There's a certain tension and ambiguity in terms of how he frames his time in prison, because while he is quite clear that he did not commit the crime he was imprisoned for, he also seems to be of the opinion that this detail does not actually matter. What matters is that the State found him guilty, and he was imprisoned for it. Furthermore, he himself claims to have committed a transgression, even if his own story would suggest otherwise.
As to the second question, I don't think the it's as simple as that the Misfit kills because he enjoys killing (I think his psychological crisis runs far deeper than that). Rather, I would say that at the core of his criminality is a deep and profound nihilism. We see this in his conversation with the grandmother, as it regards to the Christian message. As he tells her, if the Christian message amounts to a lie,

then it's nothing for you to do but enjoy the few minutes you got left the best way you can—by killing somebody or burning down his house or doing some other meanness to him.

Note, however, that he isn't simply saying that he murders because he wants to murder people. He's actually exhibiting a kind of moral insanity, where distinctions between right and wrong have been rendered meaningless. Ultimately, he kills because he recognizes no moral reasoning that justifies doing otherwise. He views this in terms of a dilemma: either Christianity must be true, in which case one must follow Christ and live as a moral actor within a moral universe, or Christianity must be false, in which case anything goes, as both moral and immoral actions do not have any value.

The Misfit's life has been pretty tragic. He says that he never was a bad boy as a child. One day, he was locked up in the penitentiary, but he knows he didn't commit any crime. He recalls,

It was a head-doctor at the penitentiary said what I had done was kill my daddy but I known that for a lie. My daddy died in nineteen ought nineteen of the epidemic flu and I never had a thing to do with it.

In other words, he was accused of murdering his father, but his father had actually been the victim of a terrible flu epidemic. Once the Misfit was put in prison, he was surrounded by walls, and despite all his thinking, he could never come up with what he had done to deserve it.
The Misfit goes on to say that the authorities "could prove [he] had committed [a crime] because they had the papers on [him]. Of course . . . they never shown [him] the papers." He says that he calls himself the Misfit because he "can't make what all [he's] done wrong fit what all [he's] gone through in punishment." He claims that there is "no pleasure but meanness," but he has an intelligence and even vulnerability to him that makes it seem as though this is not the real, or at least not the the only, reason he kills. It seems as though he's now trying to deserve or justify the punishment he received. He was a good kid that was falsely accused and got locked up, his life ruined at such a young age. Now, he does terrible things to make the crime fit the punishment he endured.