Let's see how he is described initially:
Bradley Headstone, in his decent black coat and waistcoat, and decent white shirt, and decent formal black tie, and decent pantaloons of pepper and salt, with his decent silver watch in his pocket and its decent hair-guard round his neck, looked a thoroughly decent young man of six-and-twenty. He was never seen in any other dress, and yet there was a certain stiffness in his manner of wearing this, as if there were a want of adaptation between him and it, recalling some mechanics in their holiday clothes. He had acquired mechanically a great store of teacher's knowledge. He could do mental arithmetic mechanically, sing at sight mechanically, blow various wind instruments mechanically, even play the great church organ mechanically. From his early childhood up, his mind had been a place of mechanical stowage. The arrangement of his wholesale warehouse, so that it might be always ready to meet the demands of retail dealers history here, geography there, astronomy to the right, political economy to the left—natural history, the physical sciences, figures, music, the lower mathematics, and what not, all in their several places—this care had imparted to his countenance a look of care; while the habit of questioning and being questioned had given him a suspicious manner, or a manner that would be better described as one of lying in wait. There was a kind of settled trouble in the face. It was the face belonging to a naturally slow or inattentive intellect that had toiled hard to get what it had won, and that had to hold it now that it was gotten. He always seemed to be uneasy lest anything should be missing from his mental warehouse, and taking stock to assure himself.
Suppression of so much to make room for so much, had given him a constrained manner, over and above. Yet there was enough of what was animal, and of what was fiery (though smouldering), still visible in him, to suggest that if young Bradley Headstone, when a pauper lad, had chanced to be told off for the sea, he would not have been the last man in a ship's crew. Regarding that origin of his, he was proud, moody, and sullen, desiring it to be forgotten. And few people knew of it.
He is perfectly proper and polite. But he makes the mistake of accompanying his star pupil, Charley Hexam, on a trip to Hexam's sister, Lizzie. And thus begins his obsession. A triangle of sorts arises, in which Headstone sees himself in contention with another obsessed with Lizzie: a "gentleman" named Eugene Wrayburn.
Wrayburn, unlike Headstone, was brought up in wealth. Wrayburn is totally useless, with a useless education and useless professional designation, a barrister with no business. Wrayburn takes it all very lightly. Lizzie finds herself fascinated with Wrayburn, but realizes it would not end well -- she's very low-class, and at best Wrayburn would use her for his own amusement before discarding her. She refuses Headstone outright, but she would never have accepted him even if Wrayburn had not been in the case. Headstone is terrifying:
'You know what I am going to say. I love you. What other men may mean when they use that expression, I cannot tell; what I mean is, that I am under the influence of some tremendous attraction which I have resisted in vain, and which overmasters me. You could draw me to fire, you could draw me to water, you could draw me to the gallows, you could draw me to any death, you could draw me to anything I have most avoided, you could draw me to any exposure and disgrace. This and the confusion of my thoughts, so that I am fit for nothing, is what I mean by your being the ruin of me. But if you would return a favourable answer to my offer of myself in marriage, you could draw me to any good—every good—with equal force. My circumstances are quite easy, and you would want for nothing. My reputation stands quite high, and would be a shield for yours. If you saw me at my work, able to do it well and respected in it, you might even come to take a sort of pride in me;—I would try hard that you should. Whatever considerations I may have thought of against this offer, I have conquered, and I make it with all my heart. Your brother favours me to the utmost, and it is likely that we might live and work together; anyhow, it is certain that he would have my best influence and support. I don't know what I could say more if I tried. I might only weaken what is ill enough said as it is. I only add that if it is any claim on you to be in earnest, I am in thorough earnest, dreadful earnest.'
The powdered mortar from under the stone at which he wrenched, rattled on the pavement to confirm his words.
'Stop! I implore you, before you answer me, to walk round this place once more. It will give you a minute's time to think, and me a minute's time to get some fortitude together.'
Again she yielded to the entreaty, and again they came back to the same place, and again he worked at the stone.
'Is it,' he said, with his attention apparently engrossed by it, 'yes, or no?'
'Mr Headstone, I thank you sincerely, I thank you gratefully, and hope you may find a worthy wife before long and be very happy. But it is no.'
'Is no short time necessary for reflection; no weeks or days?' he asked, in the same half-suffocated way.
'Are you quite decided, and is there no chance of any change in my favour?'
'I am quite decided, Mr Headstone, and I am bound to answer I am certain there is none.'
'Then,' said he, suddenly changing his tone and turning to her, and bringing his clenched hand down upon the stone with a force that laid the knuckles raw and bleeding; 'then I hope that I may never kill him!'
The dark look of hatred and revenge with which the words broke from his livid lips, and with which he stood holding out his smeared hand as if it held some weapon and had just struck a mortal blow, made her so afraid of him that she turned to run away. But he caught her by the arm.
'Mr Headstone, let me go. Mr Headstone, I must call for help!'
'It is I who should call for help,' he said; 'you don't know yet how much I need it.'
Lizzie flees London soon after, going into hiding with the help of the angelic Mr. Riah. That thread of the plot devolves into Wrayburn attempting to locate Lizzie. During this period, Headstone stalks Wrayburn, following him when possible. Wrayburn sees this and deliberately baits Headstone by doubling back on his path so that they cross, all the while Wrayburn is commenting on the mental agony the headmaster is undergoing. Wrayburn does eventually find Lizzie's location and tracks her down to a small factory town near a river.
Needless to say, Headstone follows, tries to kill Wrayburn by beating him near to death, and throws him into the river.
Luckily for Wrayburn and Lizzie, she happens to be in the vicinity and has had quite the experience in dragging bodies from the river (which is how the novel starts). She rescues Eugene, and they get married while he is on his deathbed (or so he thinks).
Headstone has literal fits of rage through the book, with two notable situations -- first, at a railroad station, when he hears Lizzie and Wrayburn are married, and second when Charley Hexam realizes that he's implicated in the headmaster's actions and throws off all association with his former mentor.
Then Headstone finds out the man he tried to implicate in the attack, a lock-keeper named Roger Riderhood, has tracked him down and has proof that he, Headstone, perpetrated the attack. Riderhood makes the mistake of trying to blackmail a man with a murderer's heart:
'O! I'm a going on. Don't you fear but I'll go on full-fast enough for you, and fur enough for you, without your telling. Look here, Bradley Headstone, Master. You might have split the T'other governor to chips and wedges, without my caring, except that I might have come upon you for a glass or so now and then. Else why have to do with you at all? But when you copied my clothes, and when you copied my neckhankercher, and when you shook blood upon me after you had done the trick, you did wot I'll be paid for and paid heavy for. If it come to be throw'd upon you, you was to be ready to throw it upon me, was you? Where else but in Plashwater Weir Mill Lock was there a man dressed according as described? Where else but in Plashwater Weir Mill Lock was there a man as had had words with him coming through in his boat? Look at the Lock-keeper in Plashwater Weir Mill Lock, in them same answering clothes and with that same answering red neckhankercher, and see whether his clothes happens to be bloody or not. Yes, they do happen to be bloody. Ah, you sly devil!'
Bradley, very white, sat looking at him in silence.
'But two could play at your game,' said Riderhood, snapping his fingers at him half a dozen times, 'and I played it long ago; long afore you tried your clumsy hand at it; in days when you hadn't begun croaking your lecters or what not in your school. I know to a figure how you done it. Where you stole away, I could steal away arter you, and do it knowinger than you. I know how you come away from London in your own clothes, and where you changed your clothes, and hid your clothes. I see you with my own eyes take your own clothes from their hiding-place among them felled trees, and take a dip in the river to account for your dressing yourself, to any one as might come by. I see you rise up Bradley Headstone, Master, where you sat down Bargeman. I see you pitch your Bargeman's bundle into the river. I hooked your Bargeman's bundle out of the river. I've got your Bargeman's clothes, tore this way and that way with the scuffle, stained green with the grass, and spattered all over with what bust from the blows. I've got them, and I've got you. I don't care a curse for the T'other governor, alive or dead, but I care a many curses for my own self. And as you laid your plots agin me and was a sly devil agin me, I'll be paid for it—I'll be paid for it—I'll be paid for it—till I've drained you dry!'
I believe Riderhood thinks himself safe, because he also is a murderer at heart (this is a key to the "central" plot of the novel... but the Lizzie/Headstone/Wrayburn triangle is far more interesting than the supposed central story.) Did I say at heart? Riderhood actually murdered at least one person mentioned in the novel, and it's pretty clear that he has been a serial murderer of sailors.
What then occurs is one of the more satisfying end for a pair of villains - both from poor backgrounds, both murderers, but one taking the "respectable" path and the other always being the lowest among the low:
Without taking the least notice, Bradley leaned his body against a post, in a resting attitude, and there rested with his eyes cast down. 'Being brought here,' said Riderhood, gruffly, 'I'll turn it to some use by changing my gates.' With a rattle and a rush of water, he then swung-to the lock gates that were standing open, before opening the others. So, both sets of gates were, for the moment, closed.
'You'd better by far be reasonable, Bradley Headstone, Master,' said Riderhood, passing him, 'or I'll drain you all the dryer for it, when we do settle.—Ah! Would you!'
Bradley had caught him round the body. He seemed to be girdled with an iron ring. They were on the brink of the Lock, about midway between the two sets of gates.
'Let go!' said Riderhood, 'or I'll get my knife out and slash you wherever I can cut you. Let go!'
Bradley was drawing to the Lock-edge. Riderhood was drawing away from it. It was a strong grapple, and a fierce struggle, arm and leg. Bradley got him round, with his back to the Lock, and still worked him backward.
'Let go!' said Riderhood. 'Stop! What are you trying at? You can't drown Me. Ain't I told you that the man as has come through drowning can never be drowned? I can't be drowned.'
'I can be!' returned Bradley, in a desperate, clenched voice. 'I am resolved to be. I'll hold you living, and I'll hold you dead. Come down!'
Riderhood went over into the smooth pit, backward, and Bradley Headstone upon him. When the two were found, lying under the ooze and scum behind one of the rotting gates, Riderhood's hold had relaxed, probably in falling, and his eyes were staring upward. But, he was girdled still with Bradley's iron ring, and the rivets of the iron ring held tight.
What makes Headstone so terrifying, compared to out-and-out criminals like Riderhood, is his veneer of respectability. Miss Peecher, the headmistress of the sister school to Headstone's respectable school for boys, thinks Headstone to be of perfect character, as Headstone has all the outward trappings of respectability: proper manners, proper education, proper dress. It would be so suitable for headmaster and headmistress to marry. She has no idea as to the psychotic rage within.
Riderhood's villainy, though, for all his ingratiating manner, is all on the surface. He's an outcast even in the fairly low society that Lizzie finds herself in -- nobody trusts Riderhood as he has proven so many times that he's untrustworthy. He's even been in jail for stealing from a sailor (and, as noted, there is an intimation that Riderhood has killed many men in trying to steal their possessions -- and an outright accusation of a specific man being murdered by Riderhood.) One expects violence from Riderhood, but there actually is very little. Riderhood doesn't attack Wrayburn, though he has a reason to hate Eugene. Riderhood is under control of himself - he kills only for material gain. People understand that, and don't find that concept all that scary -- especially since Riderhood very rationally targets men who people will not miss (sailors on shore leave).
Headstone, on the other hand, has been trying to keep his passions leashed, and when he unravels he unravels spectacularly. His violence is uncontrolled and frightening. As a portrait of a man whose emotional control loses a grip on his attempt at respectability in the face of sexual obsession, Headstone is engrossing and one of the best villainous characterizations in Dickens.