This unfortunate group is full of folks that are just a wee bit mentally unstable. Actually, we're just being polite. These people all went totally nuts at some point and did pretty crazy things. Some might debate whether Cassandra belongs in this clique. All her friends and family definitely thought she did. She was cursed so that nobody would believe any of her predictions for the future. So every time she told people what was about to happen, they said she was nuts. Sadly, in some versions in her story, this terrible curse actually did drive her insane.
Ajax joined this clique when Odysseus was awarded Achilles' armor over him. It made the big guy so mad that he slaughtered a ton of sheep, thinking they were his former Greek buddies. Though in some versions, the madness was caused by Athena, in all tellings of the tale Ajax was so ashamed after his bout of sheep-slaying that he committed suicide.
Like in some versions of Ajax's story, Heracles was driven crazy by a goddess. In Heracles' case, Hera was the one responsible for pushing him over the edge. Heracles' madness was particularly sad because it drove him to kill his own children. To atone, Heracles had to set out on his famous Twelve Labors.
There are plenty of different versions of this dude's story, and none of them are particularly cheery. Basically, Lycurgus was the king of Thrace, and he made the pretty terrible mistake of ticking off Dionysus, the god of wine and madness. When King Lycurgus tried to kick Dionysus out of Thrace, the god got swift revenge by making the king go crazy and kill his own wife and children. In some versions, mad Lycurgus then takes an axe and chops off his own feet or legs. In other versions, he's torn apart by his own horses.
Like Lycurgus, King Pentheus of Thebes also managed to get on Dionysus' bad side. When Pentheus forbade anyone from worshiping the god of wine, Dionysus caused all the women in town, including Pentheus' mother Agave, to go crazy, run into the mountains, and take part in wild wine-fueled Bacchanals all day. Dionysus then put Pentheus under a spell, causing him to disguise himself as a woman and go spy on the Bacchants. For his final revenge, Dionysus caused his followers to see the king and rip him limb from limb. The deathblow was delivered by Agave, who ripped her son's head from his body. For all the gory details, check out The Bacchae by Euripides.