The word “infernal” means something relating to Hell or the underworld. It comes from the word “inferno,” an old-fashioned word for “fire.”
It's not hard to see, then, why Satan in Paradise Lost should be described as “infernal.” After being cast down from Heaven by God, Satan the fallen angel has been consigned to the fiery flames of Hell, which is to be his new kingdom. Just as Heaven is God's kingdom, so Hell is Satan's. He gets to rule over this wretched place, with its burning lake of fire and grim, brooding atmosphere.
Although Satan tries his best to put a brave face on it—he states that it is “Better to reign in Hell than serve in Heaven”—he doesn't want to be confined to his new kingdom. In order to get back at God for casting him down, he has to get out and about. Only this way will he be able to corrupt God's creation by introducing sin and death into the world.
He does this by traveling to Earth, where, in the disguise of a serpent—a talking serpent, no less—he manages to trick Eve into eating the forbidden fruit of the Tree of Knowledge. Not long after, Adam follows suit, which leads to his expulsion from Paradise along with Eve. Sin and death having now been introduced into the world, the “infernal Serpent” Satan can be well pleased with his evil work.